Talk:Tensor product of modules

5303:, as you said. But it is an unjustified leap to claim this gives any "intrinsic" meaning to tensor products of elements. One "could" but that choice would constitute an unjustified original research. The approach here is to just cook up some modules satisfying the universal property; that is it! we don't give any answer to the fundamental question "what is a tensor?". In fact, the article never even defines a tensor at all (rightly so in my view). Since the exposition in the "examples" section is fairly explicit about the type of isomorphisms, I still don't think there is any issue. -- 84: 2020:. As you said, the non-commutative case seems to have a problem (Bourbaki seems to talk about the case when a module has more than two ring actions, maybe that's what is needed?). I don't think there is any issue in the commutative case: the discussion in that linked article applies if the base field is a just a commutative ring. If a module is free, even the index notion applies too (but probably not particularly interesting?) We should consider discussing more concrete cases: for example, if 74: 53: 2768:

seems to me that linear maps between finitely generated projective modules in general may be represented as matrices with elements from the underlying ring, and that elements of tensor products of arbitrary modules my be represented in this way (using potentially infinite matrices). (I'm interested specifically in the non-commutative case – the commutative case not so much.) Even if we cover only the finite basis free module case, it will be a worthwhile addition. —

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which, frustratingly, mathematicians are often very blind to, and cause all sorts of argument because of lack of distinguishing distinct concepts because they share notation or a name). In this case, we need to distinguish between the tensor product as a module, and the tensor product as a map from two modules onto a module. A bit like distinguishing the real numbers as a set and the real numbers as a ring. The two tensor products concerned are isomorphic as

5333:. What I think would be useful is to note that the isomorphism of tensor products of modules does not imply equivalence of tensor products of elements of those modules, otherwise many could make the same mistake that I did. I was using "tensor" as a convenient label; I understand that it is nonstandard (hence my "if we take ..."). My statement still holds: the tensor product of elements is, by its definition, an element of the tensor product of modules. — 6205:

with a very simple motivation for this (preferably sourced). It would be nice is the motivation also gave some insight into why this particular weakening is used, which it currently does not do. I think this property arises naturally from tensors defined as linear maps, but that is not how they are defined for modules. Bourbaki does it the other way around: they define a tensor product as a balanced product and then show isomorphism with linear maps.

4904:, tensor products of modules is not made up of tensors; they just exist independent of its elements. It's very useful to have some concrete expressions in computation, but they are not part of the definition; that's why we have the important proposition at the definition section; it is very important to emphasize that what that proposition says is not the part of the definition. My view (and probably the prevailing one) is that the canonical map 4744:). The second meaning is as the "tensor product of elements of modules", which refers to the canonical map itself. Perhaps we can sort it out by rewording (defining!) everything (several articles) so as to make this distinction clear? In this picture there is no "pair", only an object and a map, each called a tensor product in its own way. This seems to be what you are saying about the first meaning. Thus, we can say that 22: 6914:

students without a prior exposure to tensor products but that muddles the important distinction between the definition and the construction. And the distinction is important to have a logically clearer treatment of the topic; it's like you want to know what is "God" before you consider the sentence "the God exists", even if muddling the distinction is comforting and convenient.

4264:, +) is not. (Also note that the former has a natural multiplication by reals, the latter only by rationals.) How do we capture this? The codomains are isomorphic, but the maps are distinct. How does one describe this so that there is an intuition of what is happening here? Perhaps there is an example over small finite rings that behaves in the same way? (Maybe using 4868: 3829: 1399: 6348:(my construction, kinda back-to-front compared to the usual). I'm guessing that this is sufficient to define the tensor product up to isomorphism, including being a balanced product. Note that at no stage is a scalar product of the result even considered, so linearity (or even additivity) of the tensor product is not needed; it just "happens". 2554:) do generate the whole tensor product, it is enough to just map the generators (pure tensors in this case). It is clear (and in fact I just did actually check with a paper and pencil) to see the map extends by linearity. This is mentioned in the important proposition in the definition section (maybe that need to be clarified). 2716:-linear, but they still have the equivalent property of preserving the right linearity of the objects they act on, and that was what I was referring to. But it is moot: the two properties (linear and linearity-preserving) seem to be equivalent under a suitable identification, so let's stick with the standard terminology. — 3450:)? This brings to mind a picture of a multimodule that simultaneously supports left, right, top, bottom, (whatever) scalar multiplication, and that the tensor product over one or more of these simultaneously would be possible. There is presumably some compatibility constraint on these scalar multiplications, even if only 3884:-vector space. The issue is this a ring structure. In any case, the "definition" is not the best place for the discussion of this example; "examples" is. Because of the nonuniqueness of the expression it is pretty tricky to show that tensor products are not isomorphic (whence, the plenty of examples in the section). -- 6096:

Yes, indeed. I'm not firing on all cylinders. I've struck my insertion. I'd prefer to not mention inverses because the context is a general ring, hence the use of "not a left zero divisor" (simply "nonzero" would be nicer, if it suffices, because then any pair of non-commuting elements would do).

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Actually I wasn't thinking of the construction of a tensor product as a question at all; the construction is irrelevant for our discussion after all. I wanted to point out that it is not uncommon for the canonical map to be forgotten in the computation. I think you make the important point of "tensor

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If not "philosophical", then maybe "by convention". I think I get your point; that it is necessary to distinguish a pair and a mere module, but that's why I said "in practice". It seems to be that, as far as computations go in practice, the canonical map ⊗ gets forgotten/ignored. This is very similar

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where on the left there is a tensor product, on the right there is just a module. I think tensor products should work like direct sum or quotient module. There is a canonical map but that shouldn't count as part of data. in other words, in particular, when we are doing comparison, the ring over which

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We seem to be on the same page. And I agree that the matrix theoretic interpretation is interesting – it develops concepts of linear algebra directly into representations of modules (of which I formally know essentially nothing!), and this article seems to be the right place for it. Specifically, it

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I've learned to be cautious with mathematical texts: when something obvious is not written it often means that it's wrong. But here, what's the point of doing a tensor product if we don't get what we want. Why do they specifically say the image is an abelian group when one naturally wants to say it's

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Very good. I agree that the lack of a strong motivation is the biggest problem. My hope that there is an external source for it e.g. from "balanced product" seems to disappear. Another hope is in my last question: If balanced products are needed in areas different from tensor products we could steal

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With the map (the pair), is it that the domains are different? Did I say that? I said that the induced structure on the codomain is different, but I'm not sure that is a strong argument for being "incomparable" (after all, one must always specify what structure is applicable in an isomorphism). I

2677:-linear map. So "as linear map" makes sense. I'm not sure what is meant by "linearity-preserving map". It is true that there is a canonical way to map tensor elements (elements in tensor products) to linear maps, but I don't know the name for this map. Bourbaki for instance doesn't give any name. -- 1017:

The question is: should this (or whatever the correct version is) not be included in this article? Reading the article leaves the impression that the usual tensor product on vector spaces (considered as a space of linear maps) does not generalize to non-fields, and especially not to noncommutative

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It says in the lead that the result is a module ("resulting in a third module") – provided that we start with modules over a commutative ring. In the case where we start with a left- and right-module and the ring is non-commutative, your observation is accurate: we do not end up with a module, only

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I do not have the background to be able to give useful examples of tensor products over a noncommutative ring. I expect that such products could be used in geometry (e.g. in the quaterionic projective plane), where they could be used to produce general linear transformations. Even without specific

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Okay, let's get back to your original point: how should we phrase the article in this regard? All that the article needs is to make the observation that the familiar requirement of bilinearity on a tensor product of vector spaces has been reduced to the weaker property of being a balanced product,

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Sorry, I was thinking back-to-front, agreed that the map is not part of the definition. I hope I've corrected my previous post suitably. Yes, I meant ≈, but got lazy when "\equiv" gave something else (after all, people do sometimes use '=' to mean 'isomophic to'). Even so, it is a short-cut: the

5176:(never =, by the way) looks weird, but, from the module perspective, it's correct (and the isomorphism says nothing about canonical maps, rightly or not). This is also not weird if you can (and perhaps should) accept the perspective that tensor products of modules don't refer to their elements. -- 3672:

This is intriguing because it brings back the possibility of higher-order tensor products, and with it, exterior algebras; I had sort of assumed that went out of the window with noncommutativity. But now I have this picture of building tensor products out of multimodules like one builds covalently

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The "using up" works either way. No, your corrected version wasn't confusing, it is just that I am trying to order the notation so that the "used up" action of the tensor product is on the "right", adjacent to the module being operated on. This will allow those familiar with matrices just to "drop

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I disagree with "far more fundamental". This seems to be a point that the principle "Knowledge is not a textbook" is relevant. I can see constructing a tensor product as a quotient of a free module (any module is a quotient of a free module after all) is probably a more digestible approach to the

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I see the purpose of the footnote as being to say to the reader: "In case you were assuming that the familiar property applied, here's a quick illustration to show that it can't work that way." I'm not entirely happy with the clarity of my exposition, but it is start. We can try and see how it

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I thought about this and it feels this comes down to a philosophical question: what is a tensor product? Is it just some kind of module or is it a pair consisting of a module and universal map. Is an object in question a module or a pair? In the "examples" section, the attitude there is a tensor

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Unless I am misinterpreting something, I'm getting what feels to me to be a clearer picture. You use the soft term "philosophical question", but it seems to me to be a matter of definition of what one is actually referring to – one of those dual uses of terminology that bedevil mathematics (and

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When I qualified with "in some situations", I was referring to the fact that the canonical homomorphism from the tensor product to hom need not be injective. One can map tensor-product elements to linear maps; but without the canonical map being injective, one cannot do identification. (or so I

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Hello all. There is an implicit claim here which is along the lines of "if the tensor functor is not exact, minimal generating sets may not be sent to minimal generating sets, but when the functor is exact minimal generating sets DO go to minimal generating sets". Vector spaces (the functor is

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s are isomorphic as groups, but the respective maps ⊗ are not equal, and hence the pairs are not isomorphic mathematical objects. In your example, you seem to be are relying on inference of intended meaning, in this case that the '=' means isomorphism as modules. This, IMO, results from use of

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seems to me to be complete. Your extension of the idea to bimodules might not apply; this is not what we were considering (implicitly, since we were dealing with one-sided modules, not bimodules. However, your construction does give an interesting variant on the concept of a bilinear tensor

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Yargh. This one keeps playing with my brain, and Bourbaki is a bit cryptic for me on this. Am I correct in saying that w.r.t. modules, the left/right distinction for scalar multiplication is mathematically spurious (in the sense that every right scalar multiplication can be considered to be

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In short, for the noncommutative case, it seems to me that tensors build a pretty little tensor algebra, except that the order is limited to 2, and scalars do not mix with tensor products. It would be interesting to see this applied to, say, geometry (e.g. how does one define curvature of an

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As to (not) extending the trace to the noncommutative case, a demonstration of why this does not work is straightforward. IMO, we can put this statement in without much contention, even if we cannot find a source for the nonworkingness of it now. Take the simple case from of the (intended)

3638:. When an "obvious" compatibility condition is met, a module becomes a left multimodule. Simiarly, one can consider a right multimodule structure, and then, in an "obvious" way, a module that has both left and right multimodule (or just multimodule). At least this is how I understood. -- 184:

always exact in this case) are given as an example, but I've only seen this proven in an ad hoc way which doesn't mention exact functors. The claim seems plausible, but I would like to ask a more senior category theorist to double check what is written and provide confirmation. Thanks!

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product, one which does survive into the tensor product. (R,S)-, (S,T)-bimodules to have a tensor product that is bilinear (in R and T), but not in general in S, which is what we would be dealing with. I don't see that there is any contradiction with what is written in this article.

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notation that implies that the module of the tensor product is meant, not the pair. This does not mean that one can translate this back into English as "the tensor products are isomorphic"; it would better be stated as "the modules that the tensor product generates are isomorphic as

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I had exactly the same mental image as you: top action, bottom action, maybe 45-degree action, etc. But actually it's not as bad as it might appear first. The idea is simple: we simply allow more than one left or right action; what is a ring action anyway: it is a group homomorphism

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You just preempted a discourse by me with that last edit ;). I'm not sure about what you're trying to say with the rest; I fully agree, and distinctly prefer the "induced by" approach (from memory, even Bourbaki does it this way). I hope you didn't interpret my remark as sarcasm.

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to the definition of tensor product than is its universal property. Of course, the universal property is also important. But the quotient construction suffices by itself as a definition of tensor product. The universal property can then be seen as the first theorem about this

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the brackets". Right modules are like column vectors (which are more common than row vectors); matrices act on them from the left, and scalars from the right. I realize that this is at odds with Bourbaki and much of the abstract notation here on WP; we can debate the merits. —

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I still need to wrap my head around the multimodule concept. This may be getting very deep for a WP article, other than the mention of what can be achieved this way. It is way over my head, but tantalizing. I'll keep my edits to what can be shown using bimodules, for now.

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of a nontrivial commutative ring and a strictly non-commutative ring. This ring will have nonzero commutators that are all two-sided zero divisors. A bilinear form over this ring that is balanced but nontrivial can be constructed; the tensor product map is an example.

4464:-vector spaces, but they are not the same as maps. As I read the definition ("... the tensor product ... is an abelian group together with a balanced product ..."), the balanced product is a map that is an integral part of the definition: the tensor product is a pair, ( 1028:

More generally, this is true for finitely-generated projective modules (this includes, for instance, vector bundles over compact manifolds and coherent sheaves over varieties). See Bourbaki, Algebra, II.4.1. It is important, and should be added to the article.

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No, I don't, though it seems to be in use, from a general googling. The term was in the article when I got to it. I found the use of the term "middle-linear map" in other articles, so I added that here as a synonym, but that too should be checked for

5112:. But the "examples" concern modules not canonical maps. When we say two tensor products are isomorphic, they don't refer to universal properties; that should be a different kind of statements. I'm for adding more clarifying sentences. I think I get 1377:

I anticipate several further interesting cases of tensor products acting as module homorphisms. It seems to me that there are many ways to interpret the tensor product of two modules as a linear map, with bilinearity as an interesting possibility.

1968:. Unfortunately, it's about commutative algebra and doesn't cover the non-commutative case, but has some interesting stuff; e.g., Lemma 12.11. (and we should probably add some of them). In particular, currently, the article says nothing about the 6754:

for non-commutative rings I found a definition which would not support the reasoning with the non-zero commutator. Moreover, I have some doubt whether the definition in that article#section is complete. My mathematical analogy would support the

2708:-linear map (and the proof that I have in mind is extremely simple; also, I do not see Bourbaki making any such restriction). The finitely generated projective module restriction only applies if you require the identification to be bijective. 3806:-bilinear map with the middle linearity). Bourbaki is s bit old so that might be why they don't use the word "balanced product". Dummit-Foote, Abstract Algebra, a fairly standard and reliable text, uses (if I remember) "balanced product" -- 6999:(where the denotes the complex conjugate). We can still form the tensor product of the module with itself, but it clearly violates the stated result. The tensor product does match the more general result stated just below that though. — 1998:

does not seem to me to be to apply to tensor products (i.e. the duality pairing is defined, but that does not really apply to a tensor product), and even the trace (the closest think I can think of) applies only to the commutative case.

1051:-linear maps) generalize to modules?" It seems to me that the above generalization extends these to order 2 over projective modules (including the infinitely generated case, which is not an isomorphism), but not to other orders, unless 4737:. Reviewing Bourbaki, a sort of answer seems to emerge: that there are two distinct meanings of "tensor product". The first meaning is as the "tensor product of modules", which is defined as a quotient (and hence the canonical map is 547: 6185:

is a (non-trivial) zero-divisor. But for a definition of "Tensor product of modules over noncommutative rings" these intricacies need not necessarily be considered as long as an obvious danger of triviality has been anticipated.

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understand. "If" the canonical homomorphism is always injective, we should say so. Otherwise we should give a counterexample for injectivity, if doing so is not too difficult. This article is the right place for such an example.)

162:(I changed most of the HTML at the beginning to Tex.) - Sorry, I didn't realize this was such an apparently charged issue. The bilinearity axioms still render as non-Tex on my settings though. I'll let you decide what to do now. 6967:) must be assumed to make the result valid; this should really be stated upfront in the text. If one does not have this assumption, the stated result falls apart. Take, for example, a bimodule over the commutative field 3409:(more than two ring actions on a module, get it?); that's how they handle tensor products of several modules over non-commutative rings. I'm just not sure if we want to use this notion; there is probably a better way. -- 1014:. This, or something very similar to it, seems to be confirmed by various discussions on stackexchange. I may also have confused some things here, so I'd be looking for correction from those whose subject area this is. 1150: 2287: 3376:

It does but yes as you suggested it's probably not an optimal approach from the expository aspect. What is needed is to discuss the universal properties for multilinear maps and tensor product of several modules. I'm

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What I meant by a "linearity-preserving map": Just as matrices can multiply matrices and not only vectors, tensor elements can act on tensor elements as well as on modules. In this action, the cannot be described as

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I'm feeling my way, so I have a high likelihood of being wrong. We're clearly agreed that the second step (finding the trace) only applies to the commutative case. The first step seem to apply though: we know that

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It seems to be directly addressed in Bourbaki, Algebra, II.4.2 (as a specialization of the case considered in II.4.1). As I would intend to write this up to answer the question "How do tensor products (regarded as

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I've been thinking on and off about the motivation, but it seems to be missing from the sources I've browsed; they simply define a quotient that produces the balanced product property. So I'm going to give that a

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Well, I think that simply dropping the use of "bilinear" in this sense is a good solution, then: it neither sensible, notable nor referenced; it is also in direct conflict with Bourbaki's sensible use of the term.

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That's almost what I said, except that I stated it more generally: any left-invertable element is never a left zero divisor, but the converse does not hold. Some rings might contain zero divisors, but possibly

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Okay, I understand the need for a counterexample, but what it stands as a counterexample to is not clearly stated. The tensor product is, initially, only defined as an abelian group, so at first blush, as a

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is the ring of smooth functions, the contraction should be usual contraction in differential geometry (I need to brush up some diff-geo here). This case generalizes to the non-commutative case by replacing

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contains a regular noncentral element. (Maybe I missed a corner case, but that's the idea.) The determinant will have similar issues in the noncommutative case, as should the exterior algebra of a module.

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Aaah, that's very good; I didn't think of matrices at all. If you have a matrix in mind, then I completely agree that a right module is a "right" module to use. Please keep up working on the article. --

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Oops, yes, it already does. I'd prefer to have the properties present the general case first, and then the changes that result from specialization to a commutative ring. Does this make sense? —

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doesn't have a role in the isomorphism. I don't have a specific opinion on whether this is a good practice or not, but nonetheless I'm pretty sure this is the typical attitude in practice. --

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an abelian group, which is to say, addition is defined, but there is no scalar multiplication for the resulting object, which would be necessary to make it a module. This is explained under

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I've been going through some texts about tensor products and the only text I could find that goes beyond the most elementary level, besides Bourbaki, is Milne's commutative algebra notes:

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seems to be missing some of the point. At first glance, one could argue that only a one-sided module definition has been given, but in fact the natural bimodule extension (namely with

6101:-bilinear maps to triviality", even though it does force triviality of a bilinear map that has the balanced product property. For example, the action of a module's dual on the module, 4689: 1682: 1018:

rings. Yet, the category of tensor products above seems to be vast and a natural (and sensible) generalization of the more familiar (heterogeneous) tensor product of vector spaces. —

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is a right module; the idea is that one "uses up" the left action; after that he (or she?) is left with the right action, whence the my correction. The action ultimately comes from

1403:, I'd like to look for further examples of tensor products of modules acting as multilinear maps. My intuition (disclaimer: high probability of confusion due to guessing) says that 4747: 4333:

makes sense. If we insist that tensor products are pairs, this gets tricky since we have to explain what 0 means as a pair. If you adopt the "pair" picture, we cannot even compare

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making the obvious diagram commute. This can be useful (e.g. in commutative algebra) when one does not want to deal with cartesian products, which a priori have no real structure.

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Ah, I see what you're saying – fair enough. So we need to find a different wording. The phrase "may be thought of as" is problematic. Maybe "Thus, an element of a tensor product

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That is not a strong motivation. There must be some property (other than "being interesting", i.e. nontrivial) that we need for the tensor product. For example, given a right

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product of elements". I understand the concept but again that notion simply "doesn't figure" in the computation of the tensor products of modules; maybe that's related to the

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ignore the non-commutative and so perhaps it makes sense to discuss it here. In this generality something like trace map exists since it is needed for, ah, various purposes. (

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as the resultant set with the inferred structure on the set. Thus, the quotient map is not part of the definition of a quotient. We need to look at how the tensor product is

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I agree to defining the dual as you have. So, now I'm getting that we have several canonical homomorphisms, and that these exist for any type of module over the same ring

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can be written as simple products of elements of the respective modules. A few words explaining why (and how) this extends to the whole tensor product space are needed.

5275: 5059: 5039: 4571:, the group of second roots of unity (they are isomorphic since they are both cyclic of the same order). Here the isomorphism doesn't really care about the quotient map 3568: 372: 5195:

type of isomorphism should be specified. On "tensor products of modules is not made up of tensors", it seems to be that the elements of the tensor product of modules

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Yes, so I had a problem with that change. I think the idea is in some situation (f-gen projective over a comutative ring) a tensor element can be identified with a

855:, the additive group of the real numbers (and is the same as real multiplication, but for the lack of scalar multiplication on the resulting algebraic object), but 286: 6097:

Generality is actually not needed to make the point: only one counterexample is needed. Also take care: the lack of commutativity is not sufficient "to force all

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tensors, in the sense that they are retrospectively named that way (if we take any linear combination of the tensor product of elements to be a tensor). Bourbaki

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Also, I added the universal property because the article was referring to an universal property "above", which wasn't there. Then I found the universal property

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Yes, the identification of the tensor product and End is probably true for non-commutative ring as well (hopefully I will reply to your other points later). --

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the wording from "linear map" to "linearity-preserving map", my thinking may have been confused. Am I correct in saying that there is no difference between an

3234:, since it deals with the noncommutative case, which specializes to exactly the multilinear case over a single commutative ring. Essentially, given a right 463: 6631:(non-commutative:) The weakening opens the door for many non-trivial tensor products over non-commutative rings, by showing that the requirement of (full) 3979:-linearly independent of 1. I'm sorry that I do not have the detail knowledge to put this concisely, but I think you see that this is a complicated area? — 6788:

Since I do not have literature on the matter, I have to dismiss as an author of Tensor product of modules, but I will further observe it in the future. --

7022: 1413:-ary tensor products canonically map tensor products of suitable modules over the same ring into tensor products of modules, with duals used as necessary 130: 2700:

About your comment "in some situation (f-gen projective over a comutative ring)", I disagree with the restriction. I believe that it is fully general:

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I think the correct modern (post-Bourbaki) definition of the tensor product of several modules is that as the colimit of the diagram that generalize a

7017: 5739:. Since this is about a restriction on the bilinearity property only, we must avoid mention of the tensor product (not yet defined). Essentially, 4359:, as you pointed out domains are different. So, the answer in that picture would be they are "incomparable" (not the same thing as "non-same") -- 3341:

That section is not general enough (and requires updates). The general form of the associativity is already noted in the "properties" section. --

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I'm actually interested in matrix-theoretic interpretation as well. After all, the endomorphism ring of a free module can be identified with the

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The question is only about dimension, since dimension completely determines whether two vector spaces are isomorphic. What is the dimension of

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By the way, this canonical map is really canonically formulated in the sense: when it is an isomorphism, one can interpret it as saying that a

1082: 106: 2214: 4017:. That important example also shows how misleading it could be if one looks at expressions (i.e., tensors) to understand tensor products. -- 2068: 4811: 3802:

is non-commutative; I personally like "balanced product". Some text doesn't even bother to introduce a terminology (they refer instead to a

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First of all, it is standard and not problematic to define a linear map on the generating set, instead of all elements. Since pure tensors (

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in order to carry out the calculation. Obviously it does make sense, from the theoretical point of view, to compare various canonical maps

2467:{\displaystyle r\cdot \phi (x)=(r\cdot \phi )(x)=tr(x\otimes (r\cdot \phi ))=tr((x\cdot r)\otimes \phi )=\phi (x\cdot r)=\phi (x)\cdot r.} 730:

as abelian group is furnished only with the trivial scalar product. But the two are the same as sets and as abelian groups. Aren't they ?

3581: 760:-vector space. The scalar multiplication is the same under all cases under the definition given in this article (only multiplication by 6899: 6893:

AND: The beautiful construction of a tensor product as the quotient of a certain free module by the relations that must be satisfied is

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For the tensor product over the commutative ring R simply set R = S = T, thus starting with 2 R-modules and ending up with an R-module.

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On second thoughts, this does not stop us defining the trace "up to conjugation". I don't know how useful this would be, though. —

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is commutative. This suggests two diverging branches of generalization, one restrictive on the generalization with noncommutative

5873:. This is not great, though, because it does not address all possible forms of bilinearity, and the footnote should be brief. — 2485:

and that the trace can be nonzero, is a contradiction. Conclusion: the trace, as would fit the canonical map, does not exist if

6918: 4574: 4054: 1236: 1159: 4526: 2030: 6921:; the linked article gives a more concrete treatment of tensor products (and those who prefer can turn to that article). -- 4900:

perspective, but the definition of tensor products of modules actually don't involve tensor products of elements at all. By

4403:(believe me or not I've been thinking about this) Despite what I said above I think the "pair" perspective doesn't work in 4289: 6778:

If there is some ring which is non-commutative, only S survives as ring and (3) as property. The rings R and T shrink to

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the quotient map is usually not regarded as a part of the quotient; for example, that's how one can have the isomorphism

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I'm happy with the rearrangement to a remark. To me, it follows in a direct and logical fashion, is not overly verbose,

5203:

the tensor product of elements to be an element of the tensor product of modules via the canonical map: "the element of

33: 1002:

is a general ring (in particular, when it is noncommutative), provided a suitable constraint is imposed on the modules

6785:

True non-commutative rings R and T and properties (1) and (2) do not seem to support our argument with the commutator.

6639:-modularity keeps the door extremely and uncomfortably tight (proof by commutator). Furthermore pointing out that the 5169:{\displaystyle \mathbb {R} \otimes _{\mathbb {Q} }\mathbb {R} \approx \mathbb {R} \otimes _{\mathbb {R} }\mathbb {R} } 4100:

So, we would need a way of making clearer why the isomorphism is useless. Perhaps The result might be isomorphic as

6818:

Answering my own point, I have found that it is easy to construct a counterexample to my hypothesis: construct the

4955:

is something like a quotient map; it is used to formulate the universal property but that's not part of the module

6222:

Have you got a reference for the definition of "balanced product" ? Bourbaki uses the construct, but not the term.

3231: 607: 224: 6240:

The reasoning in the footnote could be extended in the way as you showed it with the commutator. In plain text ?

1687: 1030: 914:

It would seem to me that for the tensor product of finite-dimensional free modules that we have the isomorphism

4801:{\displaystyle \mathbb {R} \otimes _{\mathbb {Q} }\mathbb {R} =\mathbb {R} \otimes _{\mathbb {R} }\mathbb {R} } 21: 2663:-linear map an what I've called a linearity-preserving map (up to isomorphism)? Should I undo this change? — 756:– this allows us to think of the tensor product as a quotient of the Cartesian product, considered as a free 445: 169:, so It's duplicated now. But I don't want to delete it; I think I already messed with the article enough :) 6903: 4656: 3695: 1641: 4519:

to the situation with a quotient module; by definition, a quotient module comes with a quotient map but in

6926: 6851: 5308: 5181: 4696: 4434: 4364: 4187: 4152:. I agree that, as isomorphism, it's pretty useless. But it does constitute a counterexample to the claim 4149: 4022: 3889: 3811: 3798:

I don't know a good solution to this terminology issue. A "bilinear" could be very misleading in the case

3643: 3414: 3386: 3346: 3209: 3148: 2754: 2682: 2624: 2046: 1977: 1950: 1913: 1854: 1732: 1252: 750:

Disclaimer: not my area, but this may give some pointers; it may help to make this clearer in the article.

566: 6948: 6367:(i.e. no weakening). I prefer this approach: the tensor product of modules over a commutative ring is an 6189:

On the other hand, I never saw an example for a tensor product of modules over a noncommutative ring. --

397: 39: 6426:

Without a big further recherche I would be satisfied with the following (maybe as plain text remarks):

5933:

noncentral invertible elements. What both of us left out here is that it must not be in the center of

2143: 83: 6793: 6789: 6705: 6701: 6264: 6260: 6194: 6190: 6071: 6067: 5911: 5907: 5696: 5692: 4617: 4145: 3331:. My inclination would be to present the general case as Bourbaki does, and then to specialize it. — 1216:

is the dual module. We can then ask whether this canonical map is an isomorphism or not (it is so if

738: 734: 437: 6819: 6129:) is called bilinear, and is nontrivial no matter what the base ring is (the zero ring excepted). — 4991: 4958: 2583: 1227:-valued linear map is the same as a (linear combination) of a linear functional times some vector 105:

on Knowledge. If you would like to participate, please visit the project page, where you can join

2944:(homogeneity)). I see no problem so far: we know (from Bourbaki) that the canonical homomorphism 2746: 2017: 441: 189: 170: 89: 3111:

I think this is the matter of attitude; if there is a linear transformation from a vector space

73: 52: 5280: 3784:

likewise. I have not located other references. Should we should remove this unfortunate use of

3781: 2531: 7000: 6922: 6847: 6824: 6738: 6385: 6210: 6130: 5938: 5874: 5334: 5304: 5241: 5177: 4873: 4692: 4490: 4430: 4383: 4360: 4273: 4183: 4117: 4018: 3980: 3885: 3871: 3838: 3807: 3789: 3699: 3678: 3674: 3673:

bonded molecules out of atoms ... but this may bring with it the possibility of nonisomorphic

3639: 3455: 3437: 3410: 3382: 3360: 3342: 3332: 3205: 3190: 3144: 3102: 2769: 2750: 2717: 2678: 2664: 2639: 2620: 2519: 2503: 2491: 2042: 2000: 1973: 1946: 1928: 1909: 1850: 1823: 1728: 1545: 1379: 1248: 1064: 1036: 1019: 900: 562: 228: 6442:. (I agree with side effect, but it should be made explicit, and the well-definedness of the 5260: 5044: 5024: 4202:-vector space is significant, so this argument is relevant. But we have the conundrum that ⊗ 3544: 2619:. The example makes sense; I don't have a good feeling about the non-commutative case :) -- 2034: 1074: 212: 6225:

Can the weakening be motivated because it precludes an obvious danger of trivialization of

4140:

is the continuum (roughly because it cannot be countable and cannot be larger than that of

351: 227:. Can you suggest where changes in wording would distinguish the two cases more clearly? — 6420: 4729:

Coming from my perspective of definitions, a quotient (of modules, groups, rings, ...) is

2559: 2038: 1995: 1991: 1969: 1880: 6733:

is trivial. I can't shake the feeling that some simple math will change this to "unless

4051:

as the continuum, which doesn't feel right; I thought it'd be countably infinite, as in

1684:

given by contraction is the same as the trace map tr when it is viewed as an element of

1077:. I think, as in that article, it makes sense to first define the canonical module map: 542:{\displaystyle \operatorname {Hom} _{S}(S\otimes _{R}M,X)=\operatorname {Hom} _{R}(M,X)} 265: 4382:

think of the definition as specifying of the pair you describe, not just the module. —

2803:

suggests that I've run into a problem with my understanding of the meaning of a "right

753: 7003: 6930: 6907: 6855: 6827: 6797: 6741: 6709: 6388: 6268: 6213: 6198: 6133: 6075: 5941: 5915: 5877: 5700: 5337: 5312: 5244: 5185: 4876: 4700: 4493: 4438: 4386: 4368: 4276: 4191: 4120: 4026: 3983: 3893: 3874: 3841: 3815: 3792: 3702: 3681: 3647: 3458: 3440: 3418: 3390: 3363: 3350: 3335: 3213: 3193: 3152: 3105: 2772: 2758: 2720: 2686: 2667: 2642: 2628: 2522: 2506: 2494: 2187: 2050: 2003: 1981: 1954: 1931: 1917: 1858: 1826: 1736: 1548: 1382: 1256: 1067: 1040: 1022: 903: 742: 570: 377: 331: 311: 291: 245: 231: 193: 173: 7011: 6917:

I also want to point out that this article more or less presupposes the readers know

3447: 185: 6811: 6751: 6430:(commutative:) The weakening does not harm tensor products over commutative rings. 6416: 5736: 3834: 1073:

I'm late to the discussion, but I would mention a similar case is already noted at

996:-linear map in exactly the same way as for when the modules are vector spaces when 5257:

Yes, as a matter of notation, it is convenient to write the image of (x, y) under

3578:, as you said. Then it is not much of a leap to consider a family of left actions 6423:. (Btw, almost everything that we are doing in mathematics is kind of WP:SYNTH.) 6209:

examples, the generalization of the tensor product to modules is very natural. —

6178:

need not be a left (or right) zero non-divisor. There may even exist rings where

4148:, we reach the answer. But of course you don't need CH with more work); see also 1529:

tensor products can be composed, preserving the linearity/bilinearity properties.

6843: 5706:

It may make sense to replace of mention of quaternions with existence of a pair

5433: 3406: 3089:" to hold? If it is not injective, all this means is that more than one element 2742: 208: 102: 6758:

Let A, B, D be (R,S)-, (S,T)- resp. (R,T)-bimodules. A bilinear map φ is a map

1364:

In both cases hold in the general case, and become isomorphisms if the modules

1243:-valued differential form is just a scalar differential form times a vector in 6156: 79: 6861:

Finish defining the subject of the article before postponing that definition!

3880:

But see an example at "Examples". The problem is that they are isomorphic as

3686:

On second thoughts, perhaps the exterior algebra needs nothing more than an (

1908:, which is a bimodule. I didn't think the corrected version is confusing. -- 787:-vector space). What is significant is that the equivalences are different: 5041:

does not participate in the calculation, although it is often useful to use

3735:(although it uses the concept without naming it), and only seems to use use 3005: 2580:-linear, am I right?). To me the main issue is the module structure: since 1966: 1822:). My statement above does not take it further. Have I missed something? — 1569:

is a free module of finite rank (or even projective of finite rank) and if

6886:

defining the subject of this article before distracting the reader with a

1145:{\displaystyle {\check {E}}\otimes _{R}F\to \operatorname {Hom} _{R}(E,F)} 2282:{\displaystyle E\otimes _{R}E^{*}\to R,\,x\otimes \phi \mapsto \phi (x).} 3230:, ch. II §3.8 deals with this topic. It is clearly a generalization of 2133:{\displaystyle E^{*}\otimes _{R}E\to R,\,\phi \otimes x\mapsto \phi (x)} 1526:

there is no equivalent of the repeated tensor product over the same ring

201: 6846:

for two ring actions to several ring actions. Did I miss anything? --

4857:{\displaystyle x\otimes _{\mathbb {Q} }y\neq x\otimes _{\mathbb {R} }y} 4043:

Okay, I see what you're saying. (Unimportant aside: the example claims

4004:-vector space? infinity, but which infinity? Ask the same question for 3936:

are the same, but that is as far as it goes. Over a larger field than

1565:

I don't know if I understand the above completely. But I know that if

4897: 3574:; a right action is precisely a left action for the opposite ring of 458:

I think something (i.e., namely N) is missing in the input. There is

4182:-vector space, a fortiori, as abelian group they are isomorphic. -- 3631:{\displaystyle \pi _{i}:R\to \operatorname {End} _{\mathbb {Z} }(M)} 3052:

is required for the statement "Thus, an element of a tensor product

1395:

Okay, so now that I've added that and the world hasn't exploded yet

878:-bilinearity are not the same thing), since we get for example that 1727:

is non-commutative, then I don't know how how this thing works. --

6324:

of elements to be the most general product that retains the right

1539:-manifold?). Where to find this in say Bourbaki would be helpful. 1059:

to projective modules, and the other restrictive possibly only on

202:

Why isn't it written that the tensor product is again an R-module?

6251:-modularity, is negligible (none ?) when the ring is commutative. 6233:? Including, of course, the weakening of the module structure of 3677:, with consequences to the associativity of the tensor product. — 3526:{\displaystyle \pi :R\to \operatorname {End} _{\mathbb {Z} }(M)} 715:

So certainly the two are different in algebraic structure, i.e.

1631:{\displaystyle E^{*}\otimes _{R}E=\operatorname {End} _{R}(E)} 1523:(multiplication of a tensor product and a scalar is undefined) 1372:

are restricted to being finitely generated projective modules.

15: 1972:(paring between a module and its dual is a special case). -- 6147:

Thanx, I agree completely. Although there is always a pair (

4097:. Whichever transfinite number it is, it is probably equal.) 3143:; that would require the transformation to be injective. -- 1873:

About the definition of a dual module, I liked the old one:

6384:

I'm not sure what you're asking with your final question. —

4429:

tensor products is taken does not enter the discussion. --

6379:

of the general definition applied to the commutative case.

4607:{\displaystyle \mathbb {Z} \to \mathbb {Z} /2\mathbb {Z} } 4090:{\displaystyle {\mathfrak {c}}=\mathbb {Q} ^{\aleph _{0}}} 2140:, needs clarification. In particular, not all tensors in 1723:(if you pick a basis clearly this is the usual trace). If 1209:{\displaystyle {\check {E}}=\operatorname {Hom} _{R}(E,R)} 1063:

being commutative. This is still speculation on my part. —

6229:-bilinearity caused by the non-commutativity of the ring 4564:{\displaystyle \mathbb {Z} /2\mathbb {Z} \simeq \mu _{2}} 6681:

of the balanced product remains kind of inactive DNA in

3454:-linearity. Or am I completely missing the intention? — 6943: 6767:(2) T-right-linear in the second argument and satisfies 3855: 3833:. I'll remove it. I see there is the same problem at 3405:

Bourbaki, who know tensors the best, use the notion of

2800: 2656: 2515: 561:-algebra. (This stuff is perhaps worth mentioning.) -- 5443:

2. Change the reasoning to: The scalar multiplication

4322:{\displaystyle \mathbb {Q} \otimes _{\mathbb {Z} }G=0} 3204:

Yeah, that would be definitely the better wording. --

2797:

not quite unless the canonical homorphism is injective

1785:) is a canonical homomorphism, and consequently so is 5283: 5263: 5118: 5067: 5047: 5027: 4994: 4961: 4910: 4814: 4750: 4659: 4620: 4577: 4529: 4292: 4057: 3584: 3547: 3486: 3381:

busy and no one is doing the job for a moment :) --

2586: 2562: 2534: 2301: 2217: 2190: 2146: 2071: 1883: 1690: 1644: 1579: 1235:. This is consistent with, say, the intuition with a 1162: 1085: 466: 400: 380: 354: 334: 314: 294: 268: 248: 6725:

One can easily tighten the statement to that unless

5357:

1. Give more detail for the reasoning in footnote :

5105:{\displaystyle \otimes :M\times N\to M\otimes _{R}N} 4948:{\displaystyle \otimes :M\times N\to M\otimes _{R}N} 3135:; for me, this is not the same thing as a vector in 2747:

Ring_(mathematics)#Matrix_ring_and_endomorphism_ring

101:, a collaborative effort to improve the coverage of 6719:

I think the commutative case is adequately covered.

6696:(although I do not know literature for the latter). 6867:The definition does not prove the existence of M ⊗ 6308:. It would be natural to require a tensor product 6256:Is this relevant so far for tensor products only ? 5295: 5269: 5168: 5104: 5053: 5033: 5013: 4980: 4947: 4856: 4800: 4683: 4645: 4606: 4563: 4321: 4089: 3630: 3562: 3525: 3446:equivalent to a left scalar multiplication of the 2605: 2568: 2546: 2466: 2281: 2196: 2172: 2132: 1896: 1715: 1676: 1630: 1208: 1144: 648:Question: could this be made a bit more specific? 541: 425: 386: 366: 340: 320: 300: 280: 254: 4286:product is just a module so the calculation like 2704:tensor product element can be identified with an 2556:As for the example, I have trouble following it ( 910:Tensor product of finite-dimensional free modules 5966:being not in the center". Because if there is a 4614:. In much the same way, in the computation like 2745:(this works even for non-commutative rings; see 2033:; sheaves of rings of such are more natural but 394:-module homomorphism then there is a unique map 6399:some motivation. I fear that your construction 3101:, but this seems irrelevant to the statement. — 5937:, which I've now edited into my post above. — 4476:this yields two pairs of which the respective 1430:), and thence onto its trace (Bourbaki II.4.3) 836:under this quotient AFAICT. You can say that 225:Tensor_product_of_modules#Multilinear_mappings 6878:The tensor product can also be defined as ... 6737:is always zero". Hardly important, though. — 6677:over which the balanced product is (i.e. the 5216:which is the canonical image of the element ( 3948:is an infinite-dimensional vector space over 2041:is also interesting non-commutative case) -- 8: 5350:Property 3 differs slightly from bilinearity 4411:. How do you reconcile the computation like 2477:This, assuming a noncentral regular element 990:. That is to say, the tensor product is an 4865: 2787:Canonical map required to be injective for 1716:{\displaystyle \operatorname {End} _{R}(E)} 1438:an element of a tensor product of a module 1416:an element of a tensor product of a module 19: 5478:cannot be well-defined for noncommutative 3928:is simply wrong. The tensor product over 3902:I think that the example that claims that 3731:. I see that Bourbaki does not refer to a 242:One has the other universal property: Let 47: 6700:Any sensible reordering is acceptable. -- 6446:-scalar multiplication has to be shown.) 5886:Isn't it sufficient to find only 1 pair ( 5282: 5262: 5162: 5161: 5155: 5154: 5153: 5145: 5144: 5137: 5136: 5130: 5129: 5128: 5120: 5119: 5117: 5093: 5066: 5046: 5026: 5002: 4993: 4969: 4960: 4936: 4909: 4845: 4844: 4843: 4824: 4823: 4822: 4813: 4794: 4793: 4787: 4786: 4785: 4777: 4776: 4769: 4768: 4762: 4761: 4760: 4752: 4751: 4749: 4658: 4628: 4619: 4600: 4599: 4591: 4587: 4586: 4579: 4578: 4576: 4555: 4544: 4543: 4535: 4531: 4530: 4528: 4304: 4303: 4302: 4294: 4293: 4291: 4079: 4074: 4070: 4069: 4059: 4058: 4056: 3723:The section 'Balanced product' defines a 3610: 3609: 3608: 3589: 3583: 3546: 3505: 3504: 3503: 3485: 2594: 2585: 2561: 2533: 2300: 2235: 2225: 2216: 2189: 2161: 2151: 2145: 2086: 2076: 2070: 1888: 1882: 1695: 1689: 1659: 1649: 1643: 1610: 1594: 1584: 1578: 1182: 1164: 1163: 1161: 1118: 1102: 1087: 1086: 1084: 644:are completely different from each other. 515: 490: 471: 465: 408: 399: 379: 353: 333: 313: 293: 267: 247: 4407:if it might appear to be a good idea in 3008:, this means there do no exist distinct 4808:, and simultaneously that (in general) 4684:{\displaystyle \otimes :M\times R\to M} 3952:. In particular, the dimension of say 2651:Linearity-preserving map vs. Linear map 2250: 2104: 1677:{\displaystyle E^{*}\otimes _{R}E\to R} 1442:with its dual maps canonically into End 1420:with its dual maps canonically into End 49: 4150:Basis (linear algebra)#Related notions 3859: 2697:Okay, I've reversed that edit of mine. 1573:is commutative, then one can identify 1220:is free of finite rank, for instance). 6243:The effect of the weakening of both, 2749:) and that should be interesting. -- 7: 6729:is always a two-sided zero divisor, 3975:-linearly dependent on 1, but it is 3826: 3004:. If the canonical homomorphism is 2865:that preserve the module structure ( 1994:)? It does not seem to apply. The 1396: 1334:, there is a canonical homomorphism 1288:, there is a canonical homomorphism 608:Tensor product of modules#Definition 95:This article is within the scope of 6782:thus saving properties (1) and (2). 6438:-modularity can easily be restored 4489:-)modules". Does this make sense? — 4060: 3541:determines a linear transformation 3222:Associativity of the tensor product 870:is not the same as multiplication ( 426:{\displaystyle M\otimes _{R}N\to X} 38:It is of interest to the following 6838:Tensor products of several modules 6770:(3) φ(a s,b) = φ(a,s b) with s ∈ S 6340:′), regarded as the canonical map 4076: 2173:{\displaystyle E^{*}\otimes _{R}E} 14: 7023:Low-priority mathematics articles 5906:-bilinear maps to triviality ? -- 5232:and called the tensor product of 3698:might still be good using this. — 115:Knowledge:WikiProject Mathematics 7018:Start-Class mathematics articles 6871:N; see below for a construction. 4866: 4646:{\displaystyle M\otimes _{R}R=M} 3827: 3131:uniquely determines a vector in 3119:, I don't interpret a vector in 2986:of the tensor product to such a 2978:exists, and this just says that 1397: 118:Template:WikiProject Mathematics 82: 72: 51: 20: 6919:tensor product of vector spaces 5532:) do not map to the same coset 1237:vector-valued differential form 135:This article has been rated as 6764:(1) R-left-linear in the first 5958:I don't see the necessity of " 5508:and then the representatives ( 5083: 5014:{\displaystyle M\otimes _{R}N} 4981:{\displaystyle M\otimes _{R}N} 4926: 4675: 4583: 3780:are their respective scalars. 3625: 3619: 3601: 3557: 3551: 3520: 3514: 3496: 2617:-linearity doesn't make sense. 2606:{\displaystyle M\otimes _{R}N} 2452: 2446: 2437: 2425: 2416: 2407: 2395: 2392: 2380: 2377: 2365: 2356: 2344: 2338: 2335: 2323: 2317: 2311: 2273: 2267: 2261: 2241: 2127: 2121: 2115: 2095: 2031:ring of differential operators 1710: 1704: 1668: 1625: 1619: 1203: 1191: 1169: 1139: 1127: 1111: 1092: 668:, because the bilinear map of 536: 524: 505: 480: 417: 358: 174:13:16, 25 September 2007 (UTC) 1: 6062:. Which is the triviality of 4329:for any finite abelian group 4108:-vector spaces (only the one 3772:are elements of modules, and 3071:may be thought of as a right 1409:elements of arbitrary finite 194:20:19, 22 February 2011 (UTC) 109:and see a list of open tasks. 6363:-modularity for commutative 4132:If I remember correctly the 1435:In the noncommutative case, 904:00:08, 26 January 2015 (UTC) 743:16:32, 25 January 2015 (UTC) 4104:-vector spaces, but not as 3048:. How come injectivity of 2184:canonical map (assume that 764:is defined), and thus even 754:Tensor product § Definition 7039: 6673:is any commutative ring ⊆ 6659:can easily be upgraded to 5296:{\displaystyle x\otimes y} 4387:16:24, 30 April 2015 (UTC) 4369:13:15, 30 April 2015 (UTC) 4277:14:29, 29 April 2015 (UTC) 4233:, +) is commutative, but ⊗ 4192:12:22, 29 April 2015 (UTC) 4121:20:42, 28 April 2015 (UTC) 4027:16:20, 28 April 2015 (UTC) 3984:14:27, 28 April 2015 (UTC) 3894:12:00, 28 April 2015 (UTC) 3875:04:51, 28 April 2015 (UTC) 3854:I'm pretty convinced that 3842:15:36, 24 April 2015 (UTC) 3837:, which I'll remove too. — 3816:14:18, 24 April 2015 (UTC) 3793:05:31, 24 April 2015 (UTC) 3703:21:37, 13 April 2015 (UTC) 3682:21:33, 13 April 2015 (UTC) 3648:20:59, 13 April 2015 (UTC) 3533:. This means each element 3459:18:57, 13 April 2015 (UTC) 3441:02:48, 13 April 2015 (UTC) 3419:01:43, 13 April 2015 (UTC) 3391:01:20, 13 April 2015 (UTC) 3364:00:03, 13 April 2015 (UTC) 3351:19:16, 12 April 2015 (UTC) 3336:19:11, 12 April 2015 (UTC) 3262:, we get the isomorphisms 3214:19:15, 12 April 2015 (UTC) 3194:17:23, 12 April 2015 (UTC) 3153:16:49, 12 April 2015 (UTC) 3106:15:25, 12 April 2015 (UTC) 3093:may map to the same right 2807:-module homomorphism", or 2773:22:36, 11 April 2015 (UTC) 2759:21:58, 11 April 2015 (UTC) 2721:17:25, 11 April 2015 (UTC) 2687:15:02, 11 April 2015 (UTC) 2668:05:10, 11 April 2015 (UTC) 2613:is just an abelian group, 2547:{\displaystyle x\otimes y} 2016:Sorry, the link should be 1452:), but no trace is defined 1023:05:12, 20 March 2015 (UTC) 232:14:54, 4 August 2014 (UTC) 217:14:25, 4 August 2014 (UTC) 6931:23:01, 4 April 2018 (UTC) 6908:07:10, 4 April 2018 (UTC) 4988:. Thus, when calculating 2643:19:13, 9 April 2015 (UTC) 2629:18:33, 9 April 2015 (UTC) 2523:18:25, 9 April 2015 (UTC) 2507:17:53, 9 April 2015 (UTC) 2495:17:41, 9 April 2015 (UTC) 2051:12:46, 9 April 2015 (UTC) 2004:03:49, 9 April 2015 (UTC) 1982:22:58, 8 April 2015 (UTC) 1955:02:28, 7 April 2015 (UTC) 1932:01:55, 7 April 2015 (UTC) 1918:01:31, 7 April 2015 (UTC) 1859:02:28, 7 April 2015 (UTC) 1827:13:36, 6 April 2015 (UTC) 1737:13:13, 6 April 2015 (UTC) 1549:03:21, 6 April 2015 (UTC) 1406:Over a commutative ring, 1383:01:26, 6 April 2015 (UTC) 1257:23:19, 4 April 2015 (UTC) 1068:17:27, 4 April 2015 (UTC) 1041:13:09, 2 April 2015 (UTC) 680:is the multiplication in 571:15:42, 6 April 2015 (UTC) 238:Other universal property? 134: 67: 46: 7004:16:08, 16 May 2020 (UTC) 6856:17:46, 8 June 2015 (UTC) 6828:18:03, 8 June 2015 (UTC) 6798:16:45, 8 June 2015 (UTC) 6742:20:44, 7 June 2015 (UTC) 6710:08:15, 1 June 2015 (UTC) 6453:-scalar multiplication: 6389:22:57, 31 May 2015 (UTC) 6269:18:57, 31 May 2015 (UTC) 6214:15:00, 31 May 2015 (UTC) 6199:07:26, 31 May 2015 (UTC) 6170:≠ 0 in a noncommutative 6134:22:15, 30 May 2015 (UTC) 6076:20:20, 30 May 2015 (UTC) 5942:19:18, 30 May 2015 (UTC) 5916:15:42, 30 May 2015 (UTC) 5878:12:58, 29 May 2015 (UTC) 5842:= 0. This implies that 5728:is not in the centre of 5701:08:43, 29 May 2015 (UTC) 5270:{\displaystyle \otimes } 5054:{\displaystyle \otimes } 5034:{\displaystyle \otimes } 1990:Did you mean that link ( 141:project's priority scale 5894:) with left-invertible 5338:18:18, 4 May 2015 (UTC) 5313:16:09, 4 May 2015 (UTC) 5245:21:14, 3 May 2015 (UTC) 5186:20:38, 3 May 2015 (UTC) 4877:18:29, 3 May 2015 (UTC) 4701:17:13, 3 May 2015 (UTC) 4494:02:49, 3 May 2015 (UTC) 4439:00:45, 3 May 2015 (UTC) 3696:noncommutative geometry 3563:{\displaystyle \pi (r)} 2855:is the set of all maps 2837:. As I understand it, 2065:The trace, defined via 98:WikiProject Mathematics 6761:φ : A × B → D which is 6281:Using your numbering: 5902:in order to force all 5297: 5271: 5170: 5106: 5055: 5035: 5015: 4982: 4949: 4858: 4802: 4685: 4647: 4608: 4565: 4323: 4091: 3739:in the sense that < 3632: 3564: 3527: 2607: 2570: 2548: 2468: 2283: 2198: 2174: 2134: 1898: 1717: 1678: 1632: 1239:; locally speaking, a 1210: 1146: 984:is the dual module of 783:-vector space, not an 543: 427: 388: 368: 367:{\displaystyle M\to X} 342: 322: 302: 282: 256: 207:automatically a module 179:Verification questions 28:This article is rated 6971:with general element 6937:Sidedness of a module 6499:) of the same coset ( 5298: 5272: 5171: 5107: 5056: 5036: 5016: 4983: 4950: 4859: 4803: 4686: 4648: 4609: 4566: 4324: 4092: 3633: 3565: 3528: 2608: 2571: 2569:{\displaystyle \phi } 2549: 2469: 2284: 2199: 2175: 2135: 1899: 1897:{\displaystyle E^{*}} 1877:is a left module and 1718: 1679: 1633: 1486:) should be canonical 1211: 1147: 704:for the vector space 614:It can be shown that 544: 452:) 2014-11-23T15:05:41 428: 389: 369: 343: 323: 303: 283: 257: 6983:, defined such that 6895:far more fundamental 6812:Bilinear map#Modules 6752:Bilinear map#Modules 5281: 5261: 5224:) ... is denoted by 5116: 5065: 5045: 5025: 4992: 4959: 4908: 4812: 4748: 4657: 4618: 4575: 4527: 4290: 4146:continuum hypothesis 4055: 3582: 3545: 3484: 2655:I think that when I 2584: 2560: 2532: 2299: 2215: 2188: 2144: 2069: 1881: 1688: 1642: 1577: 1160: 1083: 606:The article says in 464: 398: 378: 352: 332: 312: 292: 266: 246: 121:mathematics articles 6820:direct sum of rings 6477:2 Representatives ( 6375:-bilinear map as a 5482:, because for some 4047:-dimensionality of 3858:gets it wrong. See 1638:and then a pairing 281:{\displaystyle M,N} 6810:The definition at 6523:Well-definedness: 6449:Definition of the 6296:, its dual module 5293: 5267: 5166: 5102: 5051: 5031: 5011: 4978: 4945: 4854: 4798: 4681: 4653:the canonical map 4643: 4604: 4561: 4319: 4165:is different from 4087: 3675:structural isomers 3628: 3560: 3523: 3380: 3232:§§ Several modules 3115:to a vector space 2795:The edit summary ( 2618: 2603: 2566: 2544: 2464: 2279: 2251: 2194: 2170: 2130: 2105: 2018:tensor contraction 1894: 1713: 1674: 1628: 1206: 1142: 539: 423: 384: 364: 338: 318: 298: 278: 262:be a ring and let 252: 90:Mathematics portal 34:content assessment 6663:-bilinearity and 6635:-bilinearity and 6434:-bilinearity and 6359:-bilinearity and 6355:Bourbaki derives 6304:-linear map from 6247:-bilinearity and 4116:-vector space). — 3378: 3127:. Each vector in 2917:(additivity) and 2555: 2197:{\displaystyle E} 2061:Trace of a tensor 1172: 1095: 978:is any ring, and 874:-bilinearity and 751: 454: 440:comment added by 387:{\displaystyle R} 341:{\displaystyle R} 321:{\displaystyle X} 301:{\displaystyle R} 255:{\displaystyle R} 155: 154: 151: 150: 147: 146: 7030: 6946: 6045: 6005: 5856: 5841: 5817: 5685: 5614: 5473: 5427: 5408: 5302: 5300: 5299: 5294: 5277:as, guess what, 5276: 5274: 5273: 5268: 5175: 5173: 5172: 5167: 5165: 5160: 5159: 5158: 5148: 5140: 5135: 5134: 5133: 5123: 5111: 5109: 5108: 5103: 5098: 5097: 5060: 5058: 5057: 5052: 5040: 5038: 5037: 5032: 5020: 5018: 5017: 5012: 5007: 5006: 4987: 4985: 4984: 4979: 4974: 4973: 4954: 4952: 4951: 4946: 4941: 4940: 4871: 4870: 4869: 4863: 4861: 4860: 4855: 4850: 4849: 4848: 4829: 4828: 4827: 4807: 4805: 4804: 4799: 4797: 4792: 4791: 4790: 4780: 4772: 4767: 4766: 4765: 4755: 4690: 4688: 4687: 4682: 4652: 4650: 4649: 4644: 4633: 4632: 4613: 4611: 4610: 4605: 4603: 4595: 4590: 4582: 4570: 4568: 4567: 4562: 4560: 4559: 4547: 4539: 4534: 4328: 4326: 4325: 4320: 4309: 4308: 4307: 4297: 4096: 4094: 4093: 4088: 4086: 4085: 4084: 4083: 4073: 4064: 4063: 3970: 3969: 3850:Over which ring? 3832: 3831: 3830: 3733:balanced product 3727:as a synonym of 3637: 3635: 3634: 3629: 3615: 3614: 3613: 3594: 3593: 3569: 3567: 3566: 3561: 3532: 3530: 3529: 3524: 3510: 3509: 3508: 3330: 3176: 3088: 3070: 3043: 3029: 3003: 2982:maps an element 2977: 2943: 2916: 2885: 2864: 2854: 2824: 2612: 2610: 2609: 2604: 2599: 2598: 2575: 2573: 2572: 2567: 2553: 2551: 2550: 2545: 2473: 2471: 2470: 2465: 2288: 2286: 2285: 2280: 2240: 2239: 2230: 2229: 2203: 2201: 2200: 2195: 2179: 2177: 2176: 2171: 2166: 2165: 2156: 2155: 2139: 2137: 2136: 2131: 2091: 2090: 2081: 2080: 2035:sheaf of modules 1903: 1901: 1900: 1895: 1893: 1892: 1722: 1720: 1719: 1714: 1700: 1699: 1683: 1681: 1680: 1675: 1664: 1663: 1654: 1653: 1637: 1635: 1634: 1629: 1615: 1614: 1599: 1598: 1589: 1588: 1402: 1401: 1400: 1215: 1213: 1212: 1207: 1187: 1186: 1174: 1173: 1165: 1151: 1149: 1148: 1143: 1123: 1122: 1107: 1106: 1097: 1096: 1088: 1075:Sheaf of modules 1033: 1013: 1007: 1001: 995: 989: 983: 977: 971: 965: 959: 953: 943: 898: 869: 854: 835: 824: 814: 778: 749: 729: 643: 628: 548: 546: 545: 540: 520: 519: 495: 494: 476: 475: 453: 434: 432: 430: 429: 424: 413: 412: 393: 391: 390: 385: 373: 371: 370: 365: 347: 345: 344: 339: 327: 325: 324: 319: 307: 305: 304: 299: 287: 285: 284: 279: 261: 259: 258: 253: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 7038: 7037: 7033: 7032: 7031: 7029: 7028: 7027: 7008: 7007: 6942: 6939: 6870: 6863: 6840: 6695: 6690: 6658: 6654: 6628: 6371:-module and an 6007: 5983: 5843: 5819: 5740: 5734: 5733: 5622: 5567: 5449: 5414: 5363: 5354:Please, either 5352: 5279: 5278: 5259: 5258: 5212: 5149: 5124: 5114: 5113: 5089: 5063: 5062: 5043: 5042: 5023: 5022: 4998: 4990: 4989: 4965: 4957: 4956: 4932: 4906: 4905: 4867: 4839: 4818: 4810: 4809: 4781: 4756: 4746: 4745: 4743: 4740: 4655: 4654: 4624: 4616: 4615: 4573: 4572: 4551: 4525: 4524: 4468:, ⊗), and over 4420: 4355: 4342: 4298: 4288: 4287: 4256: 4238: 4225: 4207: 4174: 4161: 4075: 4068: 4053: 4052: 4013: 3967: 3965: 3960:is 3, but over 3924: 3911: 3867: 3863: 3852: 3828: 3721: 3694:)-bimodule. So 3604: 3585: 3580: 3579: 3543: 3542: 3499: 3482: 3481: 3326: 3317: 3304: 3295: 3282: 3272: 3263: 3224: 3174: 3170: 3139:is a vector in 3123:as a vector in 3076: 3066: 3053: 3031: 3017: 2991: 2967: 2958: 2945: 2918: 2887: 2875: 2866: 2856: 2844: 2838: 2814: 2808: 2793: 2653: 2590: 2582: 2581: 2558: 2557: 2530: 2529: 2297: 2296: 2231: 2221: 2213: 2212: 2186: 2185: 2157: 2147: 2142: 2141: 2082: 2072: 2067: 2066: 2063: 2039:Azumaya algebra 1996:duality pairing 1992:contraction map 1970:contraction map 1884: 1879: 1878: 1817: 1803: 1794: 1776: 1767: 1691: 1686: 1685: 1655: 1645: 1640: 1639: 1606: 1590: 1580: 1575: 1574: 1515: 1498: 1473: 1464: 1447: 1425: 1398: 1352: 1343: 1306: 1297: 1178: 1158: 1157: 1114: 1098: 1081: 1080: 1031: 1009: 1003: 997: 991: 985: 979: 973: 967: 961: 955: 949: 933: 924: 918: 912: 894: 888: 879: 865: 856: 846: 837: 826: 816: 810: 797: 788: 774: 765: 725: 716: 696: 660: 639: 630: 624: 615: 604: 598: 585: 511: 486: 467: 462: 461: 435: 404: 396: 395: 376: 375: 350: 349: 330: 329: 310: 309: 290: 289: 264: 263: 244: 243: 240: 204: 181: 160: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 7036: 7034: 7026: 7025: 7020: 7010: 7009: 6949:86.179.132.186 6938: 6935: 6934: 6933: 6915: 6868: 6862: 6859: 6839: 6836: 6835: 6834: 6833: 6832: 6831: 6830: 6816: 6803: 6802: 6801: 6800: 6786: 6783: 6776: 6773: 6772: 6771: 6768: 6765: 6762: 6756: 6745: 6744: 6723: 6720: 6717: 6698: 6697: 6686: 6668: 6650: 6640: 6629: 6522: 6476: 6448: 6447: 6396: 6395: 6394: 6393: 6392: 6391: 6382: 6381: 6380: 6353: 6349: 6328:-linearity of 6286: 6274: 6273: 6272: 6271: 6257: 6253: 6252: 6241: 6238: 6223: 6217: 6216: 6206: 6145: 6144: 6143: 6142: 6141: 6140: 6139: 6138: 6137: 6136: 6085: 6084: 6083: 6082: 6081: 6080: 6079: 6078: 6066:, isn't it? -- 5949: 5948: 5947: 5946: 5945: 5944: 5921: 5920: 5919: 5918: 5881: 5880: 5735:is not a left 5727: 5726: 5689: 5688: 5687: 5686: 5619:is unequal to 5617: 5616: 5615: 5562: 5561: 5560: 5506: 5505: 5504: 5476: 5475: 5474: 5438: 5437: 5411: 5410: 5409: 5351: 5348: 5347: 5346: 5345: 5344: 5343: 5342: 5341: 5340: 5320: 5319: 5318: 5317: 5316: 5315: 5292: 5289: 5286: 5266: 5250: 5249: 5248: 5247: 5208: 5189: 5188: 5164: 5157: 5152: 5147: 5143: 5139: 5132: 5127: 5122: 5101: 5096: 5092: 5088: 5085: 5082: 5079: 5076: 5073: 5070: 5050: 5030: 5010: 5005: 5001: 4997: 4977: 4972: 4968: 4964: 4944: 4939: 4935: 4931: 4928: 4925: 4922: 4919: 4916: 4913: 4892: 4891: 4890: 4889: 4888: 4887: 4886: 4885: 4884: 4883: 4882: 4881: 4880: 4879: 4853: 4847: 4842: 4838: 4835: 4832: 4826: 4821: 4817: 4796: 4789: 4784: 4779: 4775: 4771: 4764: 4759: 4754: 4741: 4738: 4714: 4713: 4712: 4711: 4710: 4709: 4708: 4707: 4706: 4705: 4704: 4703: 4680: 4677: 4674: 4671: 4668: 4665: 4662: 4642: 4639: 4636: 4631: 4627: 4623: 4602: 4598: 4594: 4589: 4585: 4581: 4558: 4554: 4550: 4546: 4542: 4538: 4533: 4505: 4504: 4503: 4502: 4501: 4500: 4499: 4498: 4497: 4496: 4448: 4447: 4446: 4445: 4444: 4443: 4442: 4441: 4416: 4394: 4393: 4392: 4391: 4390: 4389: 4374: 4373: 4372: 4371: 4351: 4338: 4318: 4315: 4312: 4306: 4301: 4296: 4280: 4279: 4252: 4234: 4221: 4203: 4170: 4157: 4136:-dimension of 4130: 4129: 4128: 4127: 4126: 4125: 4124: 4123: 4098: 4082: 4078: 4072: 4067: 4062: 4034: 4033: 4032: 4031: 4030: 4029: 4009: 3989: 3988: 3987: 3986: 3920: 3907: 3897: 3896: 3865: 3861: 3851: 3848: 3847: 3846: 3845: 3844: 3819: 3818: 3782:EOM defines it 3720: 3717: 3716: 3715: 3714: 3713: 3712: 3711: 3710: 3709: 3708: 3707: 3706: 3705: 3684: 3659: 3658: 3657: 3656: 3655: 3654: 3653: 3652: 3651: 3650: 3627: 3624: 3621: 3618: 3612: 3607: 3603: 3600: 3597: 3592: 3588: 3559: 3556: 3553: 3550: 3522: 3519: 3516: 3513: 3507: 3502: 3498: 3495: 3492: 3489: 3468: 3467: 3466: 3465: 3464: 3463: 3462: 3461: 3443: 3426: 3425: 3424: 3423: 3422: 3421: 3398: 3397: 3396: 3395: 3394: 3393: 3369: 3368: 3367: 3366: 3354: 3353: 3322: 3313: 3300: 3291: 3278: 3268: 3223: 3220: 3219: 3218: 3217: 3216: 3199: 3198: 3197: 3196: 3166: 3156: 3155: 3062: 2963: 2954: 2871: 2840: 2810: 2792: 2785: 2784: 2783: 2782: 2781: 2780: 2779: 2778: 2777: 2776: 2775: 2762: 2761: 2739: 2728: 2727: 2726: 2725: 2724: 2723: 2709: 2698: 2690: 2689: 2652: 2649: 2648: 2647: 2646: 2645: 2632: 2631: 2602: 2597: 2593: 2589: 2565: 2543: 2540: 2537: 2518:improvement. — 2512: 2511: 2510: 2509: 2475: 2474: 2463: 2460: 2457: 2454: 2451: 2448: 2445: 2442: 2439: 2436: 2433: 2430: 2427: 2424: 2421: 2418: 2415: 2412: 2409: 2406: 2403: 2400: 2397: 2394: 2391: 2388: 2385: 2382: 2379: 2376: 2373: 2370: 2367: 2364: 2361: 2358: 2355: 2352: 2349: 2346: 2343: 2340: 2337: 2334: 2331: 2328: 2325: 2322: 2319: 2316: 2313: 2310: 2307: 2304: 2290: 2289: 2278: 2275: 2272: 2269: 2266: 2263: 2260: 2257: 2254: 2249: 2246: 2243: 2238: 2234: 2228: 2224: 2220: 2193: 2169: 2164: 2160: 2154: 2150: 2129: 2126: 2123: 2120: 2117: 2114: 2111: 2108: 2103: 2100: 2097: 2094: 2089: 2085: 2079: 2075: 2062: 2059: 2058: 2057: 2056: 2055: 2054: 2053: 2009: 2008: 2007: 2006: 1985: 1984: 1962: 1961: 1960: 1959: 1958: 1957: 1937: 1936: 1935: 1934: 1921: 1920: 1891: 1887: 1870: 1869: 1868: 1867: 1866: 1865: 1864: 1863: 1862: 1861: 1838: 1837: 1836: 1835: 1834: 1833: 1832: 1831: 1830: 1829: 1813: 1799: 1790: 1772: 1763: 1746: 1745: 1744: 1743: 1742: 1741: 1740: 1739: 1712: 1709: 1706: 1703: 1698: 1694: 1673: 1670: 1667: 1662: 1658: 1652: 1648: 1627: 1624: 1621: 1618: 1613: 1609: 1605: 1602: 1597: 1593: 1587: 1583: 1556: 1555: 1554: 1553: 1552: 1551: 1542: 1541: 1540: 1532: 1531: 1530: 1527: 1524: 1511: 1494: 1487: 1469: 1460: 1453: 1443: 1433: 1432: 1431: 1421: 1414: 1388: 1387: 1386: 1385: 1375: 1374: 1373: 1362: 1348: 1339: 1316: 1302: 1293: 1260: 1259: 1221: 1205: 1202: 1199: 1196: 1193: 1190: 1185: 1181: 1177: 1171: 1168: 1154: 1153: 1152: 1141: 1138: 1135: 1132: 1129: 1126: 1121: 1117: 1113: 1110: 1105: 1101: 1094: 1091: 1044: 1043: 1032:Sławomir Biały 946: 945: 929: 920: 911: 908: 907: 906: 890: 884: 861: 842: 806: 793: 770: 721: 692: 656: 646: 645: 635: 620: 603: 594: 581: 575: 574: 573: 551: 550: 549: 538: 535: 532: 529: 526: 523: 518: 514: 510: 507: 504: 501: 498: 493: 489: 485: 482: 479: 474: 470: 422: 419: 416: 411: 407: 403: 383: 363: 360: 357: 337: 317: 297: 277: 274: 271: 251: 239: 236: 235: 234: 203: 200: 198: 180: 177: 159: 156: 153: 152: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 7035: 7024: 7021: 7019: 7016: 7015: 7013: 7006: 7005: 7002: 6998: 6994: 6990: 6986: 6982: 6978: 6974: 6970: 6966: 6962: 6958: 6954: 6950: 6945: 6936: 6932: 6928: 6924: 6920: 6916: 6912: 6911: 6910: 6909: 6905: 6901: 6900:198.233.146.2 6896: 6891: 6889: 6885: 6881: 6879: 6874: 6872: 6860: 6858: 6857: 6853: 6849: 6845: 6837: 6829: 6826: 6821: 6817: 6813: 6809: 6808: 6807: 6806: 6805: 6804: 6799: 6795: 6791: 6787: 6784: 6781: 6777: 6774: 6769: 6766: 6763: 6760: 6759: 6757: 6753: 6749: 6748: 6747: 6746: 6743: 6740: 6736: 6732: 6728: 6724: 6721: 6718: 6714: 6713: 6712: 6711: 6707: 6703: 6693: 6689: 6684: 6680: 6676: 6672: 6666: 6662: 6657: 6653: 6648: 6644: 6638: 6634: 6630: 6626: 6623: 6619: 6615: 6611: 6608: 6604: 6601: 6597: 6593: 6589: 6586: 6582: 6578: 6575: 6571: 6567: 6563: 6560: 6556: 6552: 6548: 6544: 6541: 6537: 6533: 6530: 6526: 6520: 6517: 6513: 6509: 6505: 6502: 6498: 6495: 6491: 6487: 6483: 6480: 6475: 6471: 6468: 6464: 6460: 6456: 6452: 6445: 6441: 6437: 6433: 6429: 6428: 6427: 6424: 6422: 6418: 6414: 6410: 6406: 6402: 6390: 6387: 6383: 6378: 6374: 6370: 6366: 6362: 6358: 6354: 6350: 6347: 6343: 6339: 6335: 6331: 6327: 6323: 6319: 6315: 6311: 6307: 6303: 6299: 6295: 6291: 6287: 6283: 6282: 6280: 6279: 6278: 6277: 6276: 6275: 6270: 6266: 6262: 6258: 6255: 6254: 6250: 6246: 6242: 6239: 6236: 6232: 6228: 6224: 6221: 6220: 6219: 6218: 6215: 6212: 6207: 6203: 6202: 6201: 6200: 6196: 6192: 6187: 6184: 6181: 6177: 6173: 6169: 6165: 6161: 6158: 6154: 6150: 6135: 6132: 6128: 6124: 6120: 6116: 6112: 6108: 6104: 6100: 6095: 6094: 6093: 6092: 6091: 6090: 6089: 6088: 6087: 6086: 6077: 6073: 6069: 6065: 6061: 6057: 6053: 6049: 6043: 6039: 6035: 6031: 6027: 6023: 6019: 6015: 6011: 6003: 5999: 5995: 5991: 5987: 5981: 5977: 5973: 5969: 5965: 5961: 5957: 5956: 5955: 5954: 5953: 5952: 5951: 5950: 5943: 5940: 5936: 5932: 5927: 5926: 5925: 5924: 5923: 5922: 5917: 5913: 5909: 5905: 5901: 5897: 5893: 5889: 5885: 5884: 5883: 5882: 5879: 5876: 5872: 5868: 5864: 5860: 5854: 5850: 5846: 5839: 5835: 5831: 5827: 5823: 5815: 5811: 5807: 5803: 5799: 5795: 5791: 5787: 5783: 5779: 5775: 5771: 5767: 5763: 5759: 5755: 5751: 5747: 5743: 5738: 5731: 5725: 5721: 5717: 5713: 5709: 5705: 5704: 5703: 5702: 5698: 5694: 5684: 5680: 5677: 5673: 5669: 5665: 5661: 5657: 5653: 5649: 5645: 5641: 5637: 5633: 5629: 5625: 5621: 5620: 5618: 5613: 5609: 5606: 5602: 5598: 5594: 5590: 5586: 5582: 5578: 5574: 5570: 5566: 5565: 5563: 5558: 5554: 5550: 5546: 5542: 5538: 5534: 5533: 5531: 5527: 5523: 5519: 5515: 5511: 5507: 5503: 5499: 5496: 5495: 5493: 5489: 5485: 5481: 5477: 5472: 5468: 5464: 5460: 5456: 5452: 5448: 5447: 5446: 5445: 5444: 5441: 5435: 5431: 5425: 5421: 5417: 5413:implies that 5412: 5406: 5402: 5398: 5394: 5390: 5386: 5382: 5378: 5374: 5370: 5366: 5362: 5361: 5360: 5359: 5358: 5355: 5349: 5339: 5336: 5332: 5328: 5327: 5326: 5325: 5324: 5323: 5322: 5321: 5314: 5310: 5306: 5290: 5287: 5284: 5264: 5256: 5255: 5254: 5253: 5252: 5251: 5246: 5243: 5239: 5235: 5231: 5227: 5223: 5219: 5215: 5211: 5206: 5202: 5198: 5193: 5192: 5191: 5190: 5187: 5183: 5179: 5150: 5141: 5125: 5099: 5094: 5090: 5086: 5080: 5077: 5074: 5071: 5068: 5048: 5028: 5008: 5003: 4999: 4995: 4975: 4970: 4966: 4962: 4942: 4937: 4933: 4929: 4923: 4920: 4917: 4914: 4911: 4903: 4899: 4894: 4893: 4878: 4875: 4851: 4840: 4836: 4833: 4830: 4819: 4815: 4782: 4773: 4757: 4736: 4732: 4728: 4727: 4726: 4725: 4724: 4723: 4722: 4721: 4720: 4719: 4718: 4717: 4716: 4715: 4702: 4698: 4694: 4678: 4672: 4669: 4666: 4663: 4660: 4640: 4637: 4634: 4629: 4625: 4621: 4596: 4592: 4556: 4552: 4548: 4540: 4536: 4522: 4517: 4516: 4515: 4514: 4513: 4512: 4511: 4510: 4509: 4508: 4507: 4506: 4495: 4492: 4488: 4484: 4479: 4475: 4471: 4467: 4463: 4458: 4457: 4456: 4455: 4454: 4453: 4452: 4451: 4450: 4449: 4440: 4436: 4432: 4427: 4423: 4419: 4414: 4410: 4406: 4402: 4401: 4400: 4399: 4398: 4397: 4396: 4395: 4388: 4385: 4380: 4379: 4378: 4377: 4376: 4375: 4370: 4366: 4362: 4358: 4354: 4349: 4345: 4341: 4336: 4332: 4316: 4313: 4310: 4299: 4284: 4283: 4282: 4281: 4278: 4275: 4271: 4267: 4263: 4259: 4255: 4250: 4246: 4242: 4237: 4232: 4228: 4224: 4219: 4215: 4211: 4206: 4201: 4196: 4195: 4194: 4193: 4189: 4185: 4181: 4177: 4173: 4168: 4164: 4160: 4155: 4151: 4147: 4143: 4139: 4135: 4122: 4119: 4115: 4111: 4107: 4103: 4099: 4080: 4065: 4050: 4046: 4042: 4041: 4040: 4039: 4038: 4037: 4036: 4035: 4028: 4024: 4020: 4016: 4012: 4007: 4003: 3999: 3995: 3994: 3993: 3992: 3991: 3990: 3985: 3982: 3978: 3974: 3963: 3959: 3955: 3951: 3947: 3943: 3939: 3935: 3931: 3927: 3923: 3918: 3914: 3910: 3905: 3901: 3900: 3899: 3898: 3895: 3891: 3887: 3883: 3879: 3878: 3877: 3876: 3873: 3869: 3857: 3849: 3843: 3840: 3836: 3823: 3822: 3821: 3820: 3817: 3813: 3809: 3805: 3801: 3797: 3796: 3795: 3794: 3791: 3787: 3783: 3779: 3775: 3771: 3767: 3763: 3759: 3755: 3751: 3746: 3742: 3738: 3734: 3730: 3729:balanced form 3726: 3718: 3704: 3701: 3697: 3693: 3689: 3685: 3683: 3680: 3676: 3671: 3670: 3669: 3668: 3667: 3666: 3665: 3664: 3663: 3662: 3661: 3660: 3649: 3645: 3641: 3622: 3616: 3605: 3598: 3595: 3590: 3586: 3577: 3573: 3554: 3548: 3540: 3536: 3517: 3511: 3500: 3493: 3490: 3487: 3478: 3477: 3476: 3475: 3474: 3473: 3472: 3471: 3470: 3469: 3460: 3457: 3453: 3449: 3448:opposite ring 3444: 3442: 3439: 3434: 3433: 3432: 3431: 3430: 3429: 3428: 3427: 3420: 3416: 3412: 3408: 3404: 3403: 3402: 3401: 3400: 3399: 3392: 3388: 3384: 3375: 3374: 3373: 3372: 3371: 3370: 3365: 3362: 3358: 3357: 3356: 3355: 3352: 3348: 3344: 3340: 3339: 3338: 3337: 3334: 3329: 3325: 3320: 3316: 3311: 3307: 3303: 3298: 3294: 3289: 3285: 3281: 3276: 3271: 3266: 3261: 3257: 3253: 3249: 3245: 3241: 3237: 3233: 3229: 3221: 3215: 3211: 3207: 3203: 3202: 3201: 3200: 3195: 3192: 3188: 3184: 3180: 3173: 3169: 3164: 3160: 3159: 3158: 3157: 3154: 3150: 3146: 3142: 3138: 3134: 3130: 3126: 3122: 3118: 3114: 3110: 3109: 3108: 3107: 3104: 3100: 3096: 3092: 3087: 3083: 3079: 3074: 3069: 3065: 3060: 3056: 3051: 3047: 3042: 3038: 3034: 3028: 3024: 3020: 3015: 3011: 3007: 3002: 2998: 2994: 2989: 2985: 2981: 2975: 2971: 2966: 2961: 2957: 2952: 2948: 2942: 2938: 2934: 2930: 2926: 2922: 2914: 2910: 2906: 2902: 2898: 2894: 2890: 2886:implies that 2883: 2879: 2874: 2869: 2863: 2859: 2852: 2848: 2843: 2836: 2832: 2828: 2822: 2818: 2813: 2806: 2802: 2798: 2790: 2786: 2774: 2771: 2766: 2765: 2764: 2763: 2760: 2756: 2752: 2748: 2744: 2740: 2736: 2735: 2734: 2733: 2732: 2731: 2730: 2729: 2722: 2719: 2715: 2710: 2707: 2703: 2699: 2696: 2695: 2694: 2693: 2692: 2691: 2688: 2684: 2680: 2676: 2672: 2671: 2670: 2669: 2666: 2662: 2658: 2650: 2644: 2641: 2636: 2635: 2634: 2633: 2630: 2626: 2622: 2616: 2600: 2595: 2591: 2587: 2579: 2563: 2541: 2538: 2535: 2527: 2526: 2525: 2524: 2521: 2517: 2514:Yeah, I like 2508: 2505: 2501: 2500: 2499: 2498: 2497: 2496: 2493: 2488: 2484: 2480: 2461: 2458: 2455: 2449: 2443: 2440: 2434: 2431: 2428: 2422: 2419: 2413: 2410: 2404: 2401: 2398: 2389: 2386: 2383: 2374: 2371: 2368: 2362: 2359: 2353: 2350: 2347: 2341: 2332: 2329: 2326: 2320: 2314: 2308: 2305: 2302: 2295: 2294: 2293: 2292:We have that 2276: 2270: 2264: 2258: 2255: 2252: 2247: 2244: 2236: 2232: 2226: 2222: 2218: 2211: 2210: 2209: 2207: 2191: 2181: 2167: 2162: 2158: 2152: 2148: 2124: 2118: 2112: 2109: 2106: 2101: 2098: 2092: 2087: 2083: 2077: 2073: 2060: 2052: 2048: 2044: 2040: 2036: 2032: 2028: 2023: 2019: 2015: 2014: 2013: 2012: 2011: 2010: 2005: 2002: 1997: 1993: 1989: 1988: 1987: 1986: 1983: 1979: 1975: 1971: 1967: 1964: 1963: 1956: 1952: 1948: 1943: 1942: 1941: 1940: 1939: 1938: 1933: 1930: 1925: 1924: 1923: 1922: 1919: 1915: 1911: 1907: 1889: 1885: 1876: 1872: 1871: 1860: 1856: 1852: 1848: 1847: 1846: 1845: 1844: 1843: 1842: 1841: 1840: 1839: 1828: 1825: 1821: 1816: 1811: 1807: 1802: 1797: 1793: 1788: 1784: 1780: 1775: 1770: 1766: 1761: 1756: 1755: 1754: 1753: 1752: 1751: 1750: 1749: 1748: 1747: 1738: 1734: 1730: 1726: 1707: 1701: 1696: 1692: 1671: 1665: 1660: 1656: 1650: 1646: 1622: 1616: 1611: 1607: 1603: 1600: 1595: 1591: 1585: 1581: 1572: 1568: 1564: 1563: 1562: 1561: 1560: 1559: 1558: 1557: 1550: 1547: 1543: 1538: 1533: 1528: 1525: 1522: 1521:does not work 1518: 1514: 1509: 1505: 1501: 1497: 1492: 1488: 1485: 1481: 1477: 1472: 1467: 1463: 1458: 1454: 1451: 1446: 1441: 1437: 1436: 1434: 1429: 1424: 1419: 1415: 1412: 1408: 1407: 1405: 1404: 1394: 1393: 1392: 1391: 1390: 1389: 1384: 1381: 1376: 1371: 1367: 1363: 1360: 1356: 1351: 1346: 1342: 1337: 1333: 1329: 1325: 1321: 1317: 1314: 1310: 1305: 1300: 1296: 1291: 1287: 1283: 1279: 1275: 1271: 1270: 1268: 1264: 1263: 1262: 1261: 1258: 1254: 1250: 1246: 1242: 1238: 1234: 1230: 1226: 1222: 1219: 1200: 1197: 1194: 1188: 1183: 1179: 1175: 1166: 1155: 1136: 1133: 1130: 1124: 1119: 1115: 1108: 1103: 1099: 1089: 1079: 1078: 1076: 1072: 1071: 1070: 1069: 1066: 1062: 1058: 1054: 1050: 1042: 1038: 1034: 1027: 1026: 1025: 1024: 1021: 1015: 1012: 1006: 1000: 994: 988: 982: 976: 970: 964: 958: 952: 941: 937: 932: 927: 923: 917: 916: 915: 909: 905: 902: 897: 893: 887: 882: 877: 873: 868: 864: 859: 853: 849: 845: 840: 833: 829: 823: 819: 813: 809: 804: 800: 796: 791: 786: 782: 777: 773: 768: 763: 759: 755: 747: 746: 745: 744: 740: 736: 731: 728: 724: 719: 713: 711: 707: 703: 699: 695: 690: 685: 683: 679: 675: 671: 667: 663: 659: 654: 649: 642: 638: 633: 627: 623: 618: 613: 612: 611: 609: 601: 597: 592: 588: 584: 579: 576: 572: 568: 564: 560: 556: 552: 533: 530: 527: 521: 516: 512: 508: 502: 499: 496: 491: 487: 483: 477: 472: 468: 460: 459: 457: 456: 455: 451: 447: 443: 442:Mobius stripe 439: 420: 414: 409: 405: 401: 381: 361: 355: 335: 315: 308:-modules. If 295: 275: 272: 269: 249: 237: 233: 230: 226: 221: 220: 219: 218: 214: 210: 199: 196: 195: 191: 187: 178: 176: 175: 172: 171:Functor salad 168: 163: 157: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 6996: 6992: 6988: 6984: 6980: 6976: 6972: 6968: 6964: 6960: 6956: 6952: 6940: 6894: 6892: 6890:definition! 6887: 6883: 6882: 6877: 6875: 6866: 6864: 6841: 6779: 6734: 6730: 6726: 6699: 6691: 6687: 6682: 6678: 6674: 6670: 6667:-modularity 6664: 6660: 6655: 6651: 6646: 6642: 6636: 6632: 6624: 6621: 6617: 6613: 6609: 6606: 6602: 6599: 6595: 6591: 6587: 6584: 6580: 6576: 6573: 6569: 6565: 6561: 6558: 6554: 6550: 6546: 6542: 6539: 6535: 6531: 6528: 6524: 6518: 6515: 6511: 6507: 6503: 6500: 6496: 6493: 6489: 6485: 6481: 6478: 6473: 6469: 6466: 6462: 6458: 6454: 6450: 6443: 6440:a posteriori 6439: 6435: 6431: 6425: 6419:or at least 6412: 6408: 6404: 6400: 6397: 6376: 6372: 6368: 6364: 6360: 6356: 6345: 6341: 6337: 6333: 6329: 6325: 6321: 6317: 6313: 6309: 6305: 6301: 6297: 6293: 6289: 6248: 6244: 6234: 6230: 6226: 6188: 6182: 6179: 6175: 6171: 6167: 6163: 6159: 6152: 6148: 6146: 6126: 6122: 6118: 6114: 6110: 6106: 6102: 6098: 6063: 6059: 6055: 6051: 6047: 6041: 6037: 6033: 6029: 6025: 6021: 6017: 6013: 6009: 6006:= 0 implies 6001: 5997: 5993: 5989: 5985: 5979: 5975: 5971: 5967: 5963: 5959: 5934: 5930: 5903: 5899: 5895: 5891: 5887: 5870: 5866: 5862: 5858: 5852: 5848: 5844: 5837: 5833: 5829: 5825: 5821: 5813: 5809: 5805: 5801: 5797: 5793: 5789: 5785: 5781: 5777: 5773: 5769: 5765: 5761: 5757: 5753: 5749: 5745: 5741: 5737:zero divisor 5729: 5723: 5719: 5715: 5711: 5707: 5690: 5682: 5678: 5675: 5671: 5667: 5663: 5659: 5655: 5651: 5647: 5643: 5639: 5635: 5631: 5627: 5623: 5611: 5607: 5604: 5600: 5596: 5592: 5588: 5584: 5580: 5576: 5572: 5568: 5556: 5552: 5548: 5544: 5540: 5536: 5529: 5525: 5521: 5517: 5513: 5509: 5501: 5497: 5491: 5487: 5483: 5479: 5470: 5466: 5462: 5458: 5454: 5450: 5442: 5439: 5434:quarternions 5429: 5423: 5419: 5415: 5404: 5400: 5396: 5392: 5388: 5384: 5380: 5376: 5372: 5368: 5364: 5356: 5353: 5330: 5237: 5233: 5229: 5225: 5221: 5217: 5213: 5209: 5204: 5200: 5196: 4901: 4742:not involved 4734: 4730: 4520: 4486: 4482: 4477: 4473: 4469: 4465: 4461: 4425: 4421: 4417: 4412: 4408: 4404: 4356: 4352: 4347: 4343: 4339: 4334: 4330: 4269: 4265: 4261: 4257: 4253: 4248: 4244: 4240: 4235: 4230: 4226: 4222: 4217: 4213: 4209: 4204: 4199: 4179: 4175: 4171: 4166: 4162: 4158: 4153: 4141: 4137: 4133: 4131: 4113: 4109: 4105: 4101: 4048: 4044: 4014: 4010: 4005: 4001: 3997: 3976: 3972: 3961: 3957: 3953: 3949: 3945: 3941: 3937: 3933: 3929: 3925: 3921: 3916: 3912: 3908: 3903: 3881: 3853: 3835:Bilinear map 3803: 3799: 3786:bilinear map 3785: 3777: 3773: 3769: 3765: 3761: 3757: 3753: 3749: 3744: 3740: 3737:bilinear map 3736: 3732: 3728: 3725:bilinear map 3724: 3722: 3719:Bilinear map 3691: 3687: 3575: 3571: 3538: 3534: 3451: 3327: 3323: 3318: 3314: 3309: 3305: 3301: 3296: 3292: 3287: 3283: 3279: 3274: 3269: 3264: 3259: 3255: 3251: 3247: 3243: 3239: 3235: 3227: 3225: 3186: 3182: 3181:-linear map 3178: 3171: 3167: 3162: 3140: 3136: 3132: 3128: 3124: 3120: 3116: 3112: 3098: 3097:-linear map 3094: 3090: 3085: 3081: 3077: 3075:-linear map 3072: 3067: 3063: 3058: 3054: 3049: 3045: 3040: 3036: 3032: 3026: 3022: 3018: 3016:′ such that 3013: 3009: 3000: 2996: 2992: 2987: 2983: 2979: 2973: 2969: 2964: 2959: 2955: 2950: 2946: 2940: 2936: 2932: 2928: 2924: 2920: 2912: 2908: 2904: 2900: 2896: 2892: 2888: 2881: 2877: 2872: 2867: 2861: 2857: 2850: 2846: 2841: 2834: 2830: 2826: 2820: 2816: 2811: 2804: 2796: 2794: 2788: 2713: 2705: 2701: 2674: 2660: 2654: 2614: 2577: 2513: 2486: 2482: 2478: 2476: 2291: 2205: 2182: 2064: 2026: 2021: 1905: 1874: 1819: 1814: 1809: 1805: 1800: 1795: 1791: 1786: 1782: 1778: 1773: 1768: 1764: 1759: 1724: 1570: 1566: 1536: 1520: 1516: 1512: 1507: 1503: 1499: 1495: 1490: 1483: 1479: 1475: 1470: 1465: 1461: 1456: 1449: 1444: 1439: 1427: 1422: 1417: 1410: 1369: 1365: 1358: 1354: 1349: 1344: 1340: 1335: 1331: 1327: 1323: 1319: 1318:Given right 1312: 1308: 1303: 1298: 1294: 1289: 1285: 1281: 1277: 1273: 1266: 1244: 1240: 1232: 1228: 1224: 1217: 1060: 1056: 1052: 1048: 1045: 1016: 1010: 1004: 998: 992: 986: 980: 974: 968: 962: 960:-module and 956: 950: 947: 939: 935: 930: 925: 921: 913: 895: 891: 885: 880: 875: 871: 866: 862: 857: 851: 847: 843: 838: 831: 827: 821: 817: 811: 807: 802: 798: 794: 789: 784: 780: 775: 771: 766: 761: 757: 732: 726: 722: 717: 714: 709: 705: 701: 697: 693: 688: 686: 681: 677: 673: 669: 665: 661: 657: 652: 650: 647: 640: 636: 631: 625: 621: 616: 605: 599: 595: 590: 586: 582: 577: 558: 554: 436:— Preceding 348:-module and 241: 205: 197: 182: 166: 164: 161: 158:Tex vs. HTML 137:Low-priority 136: 96: 62:Low‑priority 40:WikiProjects 6898:definition. 6844:coequalizer 6750:Looking up 6377:side effect 6300:is a right 6285:notability. 5982:) = 1 then 4739:"forgotten" 3944:the same. 3407:multimodule 3254:and a left 2791:-linearity? 2743:matrix ring 2204:is a right 1272:Given left 954:is a right 112:Mathematics 103:mathematics 59:Mathematics 30:Start-class 7012:Categories 6790:Nomen4Omen 6755:following: 6702:Nomen4Omen 6261:Nomen4Omen 6191:Nomen4Omen 6157:commutator 6068:Nomen4Omen 5908:Nomen4Omen 5718:for which 5693:Nomen4Omen 5329:No issue, 4902:definition 4485:-(or even 3250:-bimodule 3226:Bourbaki, 3175:may act as 2825:for right 2208:-module): 966:is a left 815:only when 735:Nomen4Omen 651:Certainly 328:is a left 6944:this edit 6641:weakened 5494:there is 4472:and over 3964:it is ∞. 3856:this edit 3006:injective 2829:-modules 2801:this edit 2576:is right 1326:and left 1280:and left 972:-module, 687:But also 6645:-module 6421:WP:SYNTH 6292:-module 6046:for all 5857:for all 4521:practice 4405:practice 4112:like an 3870:above. — 3764:, where 3258:-module 3238:-module 3177:a right 3044:for any 1489:the map 1455:the map 1330:-module 1322:-module 1284:-module 1276:-module 450:contribs 438:unsigned 288:be left 186:Rschwieb 7001:Quondum 6825:Quondum 6739:Quondum 6488:) and ( 6386:Quondum 6211:Quondum 6174:, this 6155:) with 6131:Quondum 5939:Quondum 5875:Quondum 5818:, thus 5520:) and ( 5432:is the 5335:Quondum 5242:Quondum 5201:defines 4874:Quondum 4735:defined 4731:defined 4491:Quondum 4384:Quondum 4274:Quondum 4118:Quondum 3981:Quondum 3872:Quondum 3864:R ≠ R ⊗ 3839:Quondum 3790:Quondum 3700:Quondum 3679:Quondum 3456:Quondum 3438:Quondum 3361:Quondum 3333:Quondum 3228:Algebra 3191:Quondum 3103:Quondum 2770:Quondum 2718:Quondum 2665:Quondum 2657:changed 2640:Quondum 2520:Quondum 2504:Quondum 2492:Quondum 2029:by the 2001:Quondum 1929:Quondum 1824:Quondum 1812:) = End 1546:Quondum 1468:→ Bilin 1380:Quondum 1065:Quondum 1020:Quondum 901:Quondum 889:1 ≠ 1 ⊗ 752:) See 229:Quondum 139:on the 6888:second 6884:Finish 6538:) = (( 6465:) := ( 6352:works. 5654:) = (( 5638:)) = ( 5583:) = (( 5564:since 5461:) := ( 5331:per se 4898:tensor 4409:theory 2799:) for 1502:→ Hom( 1156:where 948:where 557:is an 374:is an 209:Noix07 36:scale. 6716:miss. 6553:) = ( 6417:WP:OR 6180:every 5970:with 5855:) = 0 5681:)) ⊗ 5646:) ⊗ ( 5610:)) ⊗ 5428:when 5426:) = 0 4178:. As 4144:; by 4110:feels 3956:over 3860:§ R ⊗ 3760:: --> 3747:: --> 3286:) = ( 2962:→ Hom 2899:′) = 2870:∈ Hom 2702:every 1798:→ Hom 1771:→ Hom 1347:→ Hom 1301:→ Hom 1247:. -- 928:≈ Hom 779:is a 553:when 167:below 6975:and 6927:talk 6923:Taku 6904:talk 6852:talk 6848:Taku 6794:talk 6706:talk 6612:) = 6583:= (( 6265:talk 6195:talk 6121:) ↦ 6072:talk 6032:) = 6008:0 = 5912:talk 5804:) = 5788:) = 5780:) · 5768:) = 5752:) = 5697:talk 5666:) ⊗ 5662:) · 5595:) ⊗ 5591:) · 5579:) ⊗ 5571:· (( 5543:) ⊗ 5469:) ⊗ 5395:) = 5379:) = 5309:talk 5305:Taku 5240:". — 5236:and 5182:talk 5178:Taku 4697:talk 4693:Taku 4435:talk 4431:Taku 4365:talk 4361:Taku 4346:and 4272:?) — 4188:talk 4184:Taku 4023:talk 4019:Taku 3932:and 3890:talk 3886:Taku 3812:talk 3808:Taku 3776:and 3768:and 3752:< 3644:talk 3640:Taku 3415:talk 3411:Taku 3387:talk 3383:Taku 3379:lazy 3347:talk 3343:Taku 3242:, a 3210:talk 3206:Taku 3189:"? — 3149:talk 3145:Taku 3039:′ ↦ 3030:and 3012:and 2939:) ⋅ 2931:) = 2907:) + 2833:and 2755:talk 2751:Taku 2683:talk 2679:Taku 2625:talk 2621:Taku 2516:that 2047:talk 2043:Taku 1978:talk 1974:Taku 1951:talk 1947:Taku 1914:talk 1910:Taku 1855:talk 1851:Taku 1733:talk 1729:Taku 1368:and 1361:). 1253:talk 1249:Taku 1037:talk 1008:and 739:talk 629:and 567:talk 563:Taku 446:talk 213:talk 190:talk 6941:In 6669:if 6605:)⊗( 6598:= ( 6579:))⊗ 6568:= ( 6564:))⊗ 6415:is 6162::= 6113:: ( 5806:srφ 5742:rsφ 5732:and 5674:· ( 5670:= ( 5630:⊗ ( 5626:· ( 5603:· ( 5599:= ( 5551:⊗ ( 5453:· ( 5440:or 5197:are 4260:≈ ( 4229:≈ ( 4000:as 3971:is 3942:not 3940:is 3868:R ? 3788:? — 3606:End 3570:of 3537:in 3501:End 3299:) ⊗ 2839:Hom 2809:Hom 2481:of 1693:End 1608:End 1315:). 1231:in 1180:Hom 1116:Hom 919:M ⊗ 899:. — 834:= 0 825:or 589:≠ 513:Hom 469:Hom 131:Low 7014:: 6991:= 6979:∈ 6959:= 6947:, 6929:) 6906:) 6880:" 6873:" 6854:) 6796:) 6708:) 6694:) 6627:)) 6620:⊗( 6594:)⊗ 6549:)⊗ 6534:)⊗ 6527:(( 6521:) 6514:⊗( 6510:≡ 6506:)⊗ 6492:, 6484:, 6472:)⊗ 6411:→ 6407:⊗ 6403:: 6344:→ 6336:, 6320:→ 6316:⊗ 6312:: 6267:) 6259:-- 6197:) 6168:sr 6166:− 6164:rs 6151:, 6117:, 6109:× 6105:: 6074:) 6058:∈ 6054:, 6050:∈ 6040:, 6028:, 6018:sr 6016:− 6014:rs 6000:, 5990:sr 5988:− 5986:rs 5980:sr 5978:− 5976:rs 5964:sr 5962:− 5960:rs 5931:no 5914:) 5900:sr 5898:− 5896:rs 5890:, 5869:∈ 5865:, 5861:∈ 5851:, 5836:, 5826:sr 5824:− 5822:rs 5812:, 5800:, 5796:· 5790:sφ 5784:, 5776:· 5772:(( 5764:, 5762:rs 5760:· 5748:, 5724:sr 5722:− 5720:rs 5714:∈ 5710:, 5699:) 5691:-- 5658:· 5650:· 5642:· 5634:· 5587:· 5575:· 5555:· 5547:= 5539:· 5528:· 5524:, 5516:, 5512:· 5500:≠ 5490:∊ 5486:, 5465:· 5457:⊗ 5422:, 5403:, 5397:rφ 5391:· 5387:, 5375:, 5371:· 5311:) 5288:⊗ 5265:⊗ 5228:⊗ 5220:, 5184:) 5151:⊗ 5142:≈ 5126:⊗ 5091:⊗ 5084:→ 5078:× 5069:⊗ 5049:⊗ 5029:⊗ 5021:, 5000:⊗ 4967:⊗ 4934:⊗ 4927:→ 4921:× 4912:⊗ 4864:. 4841:⊗ 4834:≠ 4820:⊗ 4783:⊗ 4758:⊗ 4699:) 4676:→ 4670:× 4661:⊗ 4626:⊗ 4584:→ 4553:μ 4549:≃ 4437:) 4424:= 4367:) 4300:⊗ 4268:/6 4247:→ 4243:× 4239:: 4216:→ 4212:× 4208:: 4190:) 4077:ℵ 4025:) 3915:≈ 3892:) 3814:) 3748:= 3745:yb 3741:au 3646:) 3617:⁡ 3602:→ 3587:π 3549:π 3512:⁡ 3497:→ 3488:π 3417:) 3389:) 3349:) 3308:= 3212:) 3185:→ 3151:) 3084:→ 3080:: 3057:∈ 3035:: 3025:↦ 3021:: 2999:↦ 2995:: 2990:, 2972:, 2949:: 2927:⋅ 2915:′) 2895:+ 2880:, 2860:→ 2849:, 2819:, 2757:) 2685:) 2627:) 2592:⊗ 2564:ϕ 2539:⊗ 2456:⋅ 2444:ϕ 2432:⋅ 2423:ϕ 2414:ϕ 2411:⊗ 2402:⋅ 2375:ϕ 2372:⋅ 2363:⊗ 2333:ϕ 2330:⋅ 2309:ϕ 2306:⋅ 2265:ϕ 2262:↦ 2259:ϕ 2256:⊗ 2242:→ 2237:∗ 2223:⊗ 2159:⊗ 2153:∗ 2119:ϕ 2116:↦ 2110:⊗ 2107:ϕ 2096:→ 2084:⊗ 2078:∗ 2049:) 1980:) 1953:) 1916:) 1890:∗ 1857:) 1808:, 1781:, 1735:) 1702:⁡ 1669:→ 1657:⊗ 1651:∗ 1617:⁡ 1592:⊗ 1586:∗ 1519:) 1506:, 1482:; 1478:, 1357:, 1311:, 1269:: 1255:) 1189:⁡ 1170:ˇ 1125:⁡ 1112:→ 1100:⊗ 1093:ˇ 1039:) 938:, 850:= 830:= 820:∈ 812:ry 801:= 790:xr 741:) 733:-- 712:. 700:= 684:. 676:→ 672:× 664:= 610:: 569:) 522:⁡ 488:⊗ 478:⁡ 448:• 418:→ 406:⊗ 359:→ 215:) 192:) 6997:r 6995:⋅ 6993:x 6989:x 6987:⋅ 6985:r 6981:C 6977:r 6973:x 6969:C 6965:r 6963:⋅ 6961:x 6957:x 6955:⋅ 6953:r 6925:( 6902:( 6876:" 6869:R 6865:" 6850:( 6823:— 6792:( 6780:Z 6735:c 6731:φ 6727:c 6704:( 6692:N 6688:R 6685:⊗ 6683:M 6679:R 6675:R 6671:S 6665:S 6661:S 6656:N 6652:R 6649:⊗ 6647:M 6643:Z 6637:R 6633:R 6625:n 6622:s 6618:m 6616:( 6614:r 6610:n 6607:s 6603:r 6600:m 6596:n 6592:s 6590:) 6588:r 6585:m 6581:n 6577:s 6574:r 6572:( 6570:m 6566:n 6562:r 6559:s 6557:( 6555:m 6551:n 6547:r 6545:) 6543:s 6540:m 6536:n 6532:s 6529:m 6525:r 6519:n 6516:s 6512:m 6508:n 6504:s 6501:m 6497:n 6494:s 6490:m 6486:n 6482:s 6479:m 6474:n 6470:r 6467:m 6463:n 6461:⊗ 6459:m 6457:( 6455:r 6451:R 6444:R 6436:R 6432:R 6413:G 6409:M 6405:N 6401:φ 6373:R 6369:R 6365:R 6361:R 6357:R 6346:N 6342:M 6338:m 6334:n 6332:( 6330:φ 6326:R 6322:G 6318:M 6314:N 6310:φ 6306:M 6302:R 6298:M 6294:M 6290:R 6263:( 6249:R 6245:R 6237:. 6235:G 6231:R 6227:R 6193:( 6183:t 6176:t 6172:R 6160:t 6153:s 6149:r 6127:n 6125:( 6123:m 6119:n 6115:m 6111:N 6107:N 6103:ψ 6099:R 6070:( 6064:φ 6060:N 6056:n 6052:M 6048:m 6044:) 6042:n 6038:m 6036:( 6034:φ 6030:n 6026:m 6024:( 6022:φ 6020:) 6012:( 6010:t 6004:) 6002:n 5998:m 5996:( 5994:φ 5992:) 5984:( 5974:( 5972:t 5968:t 5935:R 5910:( 5904:R 5892:s 5888:r 5871:N 5867:n 5863:M 5859:m 5853:n 5849:m 5847:( 5845:φ 5840:) 5838:n 5834:m 5832:( 5830:φ 5828:) 5820:( 5816:) 5814:n 5810:m 5808:( 5802:n 5798:r 5794:m 5792:( 5786:n 5782:s 5778:r 5774:m 5770:φ 5766:n 5758:m 5756:( 5754:φ 5750:n 5746:m 5744:( 5730:R 5716:R 5712:s 5708:r 5695:( 5683:n 5679:s 5676:r 5672:m 5668:n 5664:s 5660:r 5656:m 5652:n 5648:s 5644:r 5640:m 5636:n 5632:s 5628:m 5624:r 5612:n 5608:r 5605:s 5601:m 5597:n 5593:r 5589:s 5585:m 5581:n 5577:s 5573:m 5569:r 5559:) 5557:n 5553:s 5549:m 5545:n 5541:s 5537:m 5535:( 5530:n 5526:s 5522:m 5518:n 5514:s 5510:m 5502:s 5498:r 5492:R 5488:s 5484:r 5480:R 5471:n 5467:r 5463:m 5459:n 5455:m 5451:r 5436:. 5430:R 5424:n 5420:m 5418:( 5416:φ 5407:) 5405:n 5401:m 5399:( 5393:n 5389:r 5385:m 5383:( 5381:φ 5377:n 5373:r 5369:m 5367:( 5365:φ 5307:( 5291:y 5285:x 5238:y 5234:x 5230:y 5226:x 5222:y 5218:x 5214:F 5210:A 5207:⊗ 5205:E 5180:( 5163:R 5156:R 5146:R 5138:R 5131:Q 5121:R 5100:N 5095:R 5087:M 5081:N 5075:M 5072:: 5009:N 5004:R 4996:M 4976:N 4971:R 4963:M 4943:N 4938:R 4930:M 4924:N 4918:M 4915:: 4872:— 4852:y 4846:R 4837:x 4831:y 4825:Q 4816:x 4795:R 4788:R 4778:R 4774:= 4770:R 4763:Q 4753:R 4695:( 4679:M 4673:R 4667:M 4664:: 4641:M 4638:= 4635:R 4630:R 4622:M 4601:Z 4597:2 4593:/ 4588:Z 4580:Z 4557:2 4545:Z 4541:2 4537:/ 4532:Z 4487:Z 4483:Q 4478:M 4474:R 4470:Q 4466:M 4462:Q 4433:( 4426:M 4422:M 4418:R 4415:⊗ 4413:R 4363:( 4357:R 4353:R 4350:⊗ 4348:R 4344:R 4340:Z 4337:⊗ 4335:R 4331:G 4317:0 4314:= 4311:G 4305:Z 4295:Q 4270:Z 4266:Z 4262:R 4258:R 4254:Q 4251:⊗ 4249:R 4245:R 4241:R 4236:Q 4231:R 4227:R 4223:R 4220:⊗ 4218:R 4214:R 4210:R 4205:R 4200:Z 4186:( 4180:Q 4176:R 4172:R 4169:⊗ 4167:R 4163:R 4159:Z 4156:⊗ 4154:R 4142:R 4138:R 4134:Q 4114:R 4106:R 4102:Q 4081:0 4071:Q 4066:= 4061:c 4049:R 4045:Q 4021:( 4015:R 4011:Q 4008:⊗ 4006:R 4002:Q 3998:R 3977:Q 3973:R 3968:2 3966:√ 3962:Q 3958:R 3954:R 3950:Q 3946:R 3938:Q 3934:Q 3930:Z 3926:R 3922:R 3919:⊗ 3917:R 3913:R 3909:Z 3906:⊗ 3904:R 3888:( 3882:Q 3866:R 3862:Z 3810:( 3804:Z 3800:R 3778:b 3774:a 3770:v 3766:u 3762:b 3758:y 3756:, 3754:u 3750:a 3743:, 3692:R 3690:, 3688:R 3642:( 3626:) 3623:M 3620:( 3611:Z 3599:R 3596:: 3591:i 3576:R 3572:M 3558:) 3555:r 3552:( 3539:R 3535:r 3521:) 3518:M 3515:( 3506:Z 3494:R 3491:: 3452:Z 3436:— 3413:( 3385:( 3345:( 3328:N 3324:S 3321:⊗ 3319:M 3315:R 3312:⊗ 3310:L 3306:N 3302:S 3297:M 3293:R 3290:⊗ 3288:L 3284:N 3280:S 3277:⊗ 3275:M 3273:( 3270:R 3267:⊗ 3265:L 3260:N 3256:S 3252:M 3248:S 3246:, 3244:R 3240:L 3236:R 3208:( 3187:F 3183:E 3179:R 3172:E 3168:R 3165:⊗ 3163:F 3147:( 3141:W 3137:V 3133:W 3129:V 3125:W 3121:V 3117:W 3113:V 3099:φ 3095:R 3091:ξ 3086:F 3082:E 3078:ξ 3073:R 3068:E 3064:R 3061:⊗ 3059:F 3055:ξ 3050:θ 3046:φ 3041:φ 3037:ξ 3033:θ 3027:φ 3023:ξ 3019:θ 3014:ξ 3010:ξ 3001:φ 2997:ξ 2993:θ 2988:φ 2984:ξ 2980:θ 2976:) 2974:F 2970:E 2968:( 2965:R 2960:E 2956:R 2953:⊗ 2951:F 2947:θ 2941:r 2937:e 2935:( 2933:φ 2929:r 2925:e 2923:( 2921:φ 2919:( 2913:e 2911:( 2909:φ 2905:e 2903:( 2901:φ 2897:e 2893:e 2891:( 2889:φ 2884:) 2882:F 2878:E 2876:( 2873:R 2868:φ 2862:F 2858:E 2853:) 2851:F 2847:E 2845:( 2842:R 2835:F 2831:E 2827:R 2823:) 2821:F 2817:E 2815:( 2812:R 2805:R 2789:R 2753:( 2714:R 2706:R 2681:( 2675:R 2661:R 2638:— 2623:( 2615:R 2601:N 2596:R 2588:M 2578:R 2542:y 2536:x 2490:— 2487:R 2483:R 2479:r 2462:. 2459:r 2453:) 2450:x 2447:( 2441:= 2438:) 2435:r 2429:x 2426:( 2420:= 2417:) 2408:) 2405:r 2399:x 2396:( 2393:( 2390:r 2387:t 2384:= 2381:) 2378:) 2369:r 2366:( 2360:x 2357:( 2354:r 2351:t 2348:= 2345:) 2342:x 2339:( 2336:) 2327:r 2324:( 2321:= 2318:) 2315:x 2312:( 2303:r 2277:. 2274:) 2271:x 2268:( 2253:x 2248:, 2245:R 2233:E 2227:R 2219:E 2206:R 2192:E 2168:E 2163:R 2149:E 2128:) 2125:x 2122:( 2113:x 2102:, 2099:R 2093:E 2088:R 2074:E 2045:( 2027:R 2022:R 1999:— 1976:( 1949:( 1912:( 1906:R 1886:E 1875:E 1853:( 1820:E 1818:( 1815:R 1810:E 1806:E 1804:( 1801:R 1796:E 1792:R 1789:⊗ 1787:E 1783:F 1779:E 1777:( 1774:R 1769:F 1765:R 1762:⊗ 1760:E 1731:( 1725:R 1711:) 1708:E 1705:( 1697:R 1672:R 1666:E 1661:R 1647:E 1626:) 1623:E 1620:( 1612:R 1604:= 1601:E 1596:R 1582:E 1571:R 1567:E 1544:— 1537:H 1517:F 1513:R 1510:⊗ 1508:E 1504:R 1500:F 1496:R 1493:⊗ 1491:E 1484:R 1480:F 1476:E 1474:( 1471:Z 1466:F 1462:R 1459:⊗ 1457:E 1450:E 1448:( 1445:R 1440:E 1428:E 1426:( 1423:R 1418:E 1411:n 1378:— 1370:F 1366:E 1359:F 1355:E 1353:( 1350:R 1345:F 1341:R 1338:⊗ 1336:E 1332:F 1328:R 1324:E 1320:R 1313:F 1309:E 1307:( 1304:R 1299:F 1295:R 1292:⊗ 1290:E 1286:F 1282:R 1278:E 1274:R 1267:R 1251:( 1245:F 1241:F 1233:F 1229:v 1225:F 1218:E 1204:) 1201:R 1198:, 1195:E 1192:( 1184:R 1176:= 1167:E 1140:) 1137:F 1134:, 1131:E 1128:( 1120:R 1109:F 1104:R 1090:E 1061:R 1057:R 1053:R 1049:R 1035:( 1011:N 1005:M 999:R 993:R 987:N 981:N 975:R 969:R 963:N 957:R 951:M 944:, 942:) 940:N 936:M 934:( 931:R 926:N 922:R 896:π 892:Z 886:Z 883:⊗ 881:π 876:R 872:Z 867:R 863:Z 860:⊗ 858:R 852:R 848:R 844:R 841:⊗ 839:R 832:y 828:x 822:Q 818:r 808:Z 805:⊗ 803:x 799:y 795:Z 792:⊗ 785:R 781:Z 776:R 772:R 769:⊗ 767:R 762:Z 758:Z 748:( 737:( 727:R 723:Z 720:⊗ 718:R 710:R 708:/ 706:R 702:R 698:R 694:R 691:⊗ 689:R 682:R 678:R 674:R 670:R 666:R 662:R 658:Z 655:⊗ 653:R 641:R 637:Z 634:⊗ 632:R 626:R 622:R 619:⊗ 617:R 602:? 600:R 596:R 593:⊗ 591:R 587:R 583:Z 580:⊗ 578:R 565:( 559:R 555:S 537:) 534:X 531:, 528:M 525:( 517:R 509:= 506:) 503:X 500:, 497:M 492:R 484:S 481:( 473:S 444:( 421:X 415:N 410:R 402:M 382:R 362:X 356:M 336:R 316:X 296:R 276:N 273:, 270:M 250:R 211:( 188:( 143:. 42::

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Index