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extend the same way to the integers. But I think it would be fair to just mention this at the top and to state that "the" for the rest of the article means "the one unique up to isomorphism". But this is a universal issue in mathematics: when is equality "equality" and when is it just "isomorphism", and do we only use the definite article in the former case? The answer obviously depends on context, which is why you even hear people talk about "the singleton set", even when such a construction is obviously not unique.
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333:. I do not understand the detail of objection particularly, perhaps we should check it first. Since we do seem to be attempting to address the article at the trivial objects of multiple types, perhaps we should do this properly, and rename it accordingly (and not simply by the most general case, which would be the singleton): sets, groups, modules, vector spaces, rings, algebras, and maybe others, as well as to give the dominant names for each. Interesting thought: fields apparently have no such
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1111:). One option might be again to consider renaming this article more inclusively to encapsulate the "trivial object" concept. Alternatively, as you suggest, another article could be created to cover this topic (and zero objects would perhaps be classified under them, along with the trivial ring and related trivial objects). To have an article entirely devoted to the trivial ring while there are other similar "trivial objects" still does not seem right, though. —
33:
1087:, almost anything which could justify a separate article, today I am not sure that redirecting it to here was a right solution. May be it is a discussion about unital structures what ought to constitute its topic, distinct from true zero objects. Should we restore the article and move the section to there?
1318:
mentioned above seems to try to address this, but partly leaves the concept in the air with statements such as "depends on the precise definition". From the diff it would seem that the trivial ring is a zero object considered as a rng, but not when considered as a ring. This seems to match with the
1195:
So, what do you think is a right thing? An article about the trivial ring without mentioning its category-theoretical properties, eh? These trivial objects are not so easy as one could imagine. Since
Rumping intervened to this complex matter (by undoing my changes), we should expect some constructive
1313:
As someone who has tried to crystalize these concepts from WP I find this rather confusing. This article seems to draw somewhat on the concept of a zero object (in category theory), which is evidently not the same as the elementary concept of a zero object that this article seems to be addressing.
1145:
It was neither polite nor practical to revert to a version of "trivial ring" criticized for absence of essential information, especially while this information is already present in a section of another article. One does not need to have good thoughts to just push the "undo" link in a web-interface.
805:
because of different notation for the identity element. I will use the original name proposal (although do not insist that it is an unlimate solution), because "trivial object" is ambigous (see my reasonings about initial objects above), "Zero object (abstract algebra)" imposes a misleading contrast
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trivial group, but the trivial subgroup of the integers may hardly be considered to be the same as the trivial subgroup of the rotations of a
Euclidean vector space (for example). Isomorphism is not equality. On the other hand, for a stand alone trivial group, like in an exact sequence, there is no
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I think this nevertheless has (plenty of) merit. It seems to get rid of the ambiguity in the title. We can make the content less technical if need be, introducing it as generally being the simplest object in any of several categories and that it is always a singleton, and then have a section for
1403:
Even for groups, there can be different ones, which are only equal up to isomorphism. The trivial multiplicative subgroup is {1} and the trivial additive subgroup is {0} (say of the integers). These are not equal in the strictest sense because 1 is not equal to 0, and also because + and * do not
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trivial ring", the argument of isomorphism seems to be compelling. In the case of a module, a vector space or an algebra, the very definition of these categories implies the semantic of the associated scalar space ("over a ring"). This association seems to remain in the definition of the
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The problem is that "zero element" does not claim itself to be an article, but a dab page, so it has either be coverted to an article, or reformatted as a dab. Meanwhile, I proceed to the move and merger of "trivial ring", because it unlikely will be opposed. Some problems expected with
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article's key points are that {0} on its own, with ordinary addition and multiplication, is a ring and that if the additive and the multiplicative identity are the same then there is only one element, things far easier to understand. You may think that these points are repeated in the
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of the category of the modules over a ring. In fact, historically, the theory of categories (especially abelian categories) was, for a large part, a generalization of the theory of modules which was needed for homological algebra. Thus the name "zero object" is derived from "zero
272:, where it is denoted simply by 0 (it appears twice in the short exact sequences). In any case one has to look on books of homological algebra to see how this module is named there. I believe "null module", but I may be influenced by the usual French denomination of "module nul".
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zero object (category theory), but that is not what this article is about. It also seems to suggest that the definition of rings requires that 1≠0. So I have to admit continued confusion, but I'm sure that between you a more easily understood presentation will emerge. —
1299:(in a category). In other words, there is no mathematical disagreement between Incnis Mrsi and me. My point is that the considerations of theory of categories have to appear after the elementary considerations. Otherwise the article could be tagged as {{tl:Technical}}.
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a less abstract one, a matrix algebra. The object concerned (a trivial module) has already been adequately defined. I felt that the mention of matrices was illustrative only. An empty matrix may be given as an example, but perhaps that should not be in the lead. —
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My impression (which may be wrong) is that Incnis Mrsi does not think that the trivial ring is an example of a zero object (algebra), but that D.Lazard thinks that it is. I think that this should be resolved before the correctness of merging can be finalized. —
1027:
Your phrase "even
Quondum" accords me more honour than I am worthy of. I can see the thrust of what you are saying in the wording used in the article. Thank you for this: I can see that I should focus more study on morphisms (and category theory generally). —
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corresponding zero object, so that despite isomorphism in other respects, the operation of scalar multiplication is incompatible when the underlying scalars are not of the same type, and it makes less sense to regard all zero objects of these types to be
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seem to me to be non-ideal (it mentions zero objects, but is not about them as such) and that in the cases mentioned, it is exactly what this article is supposed to be changed to refer to. The initial and terminal objects that don't conform (e.g. the
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matrix, or by 0×0×0 matrix. In any case there is only one empty matrix, does not matter the square one or the cubic. But assertion of the isomorphism without an exact definition of "empty matrix" is even less precise speech than my version.
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needs to be checked for standardness, but I guess it to be more used specifically as over a field and not a ring, and is thus may be a poor choice. We may be faced with many poor options (blame non-standardization of terminology). —
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It is a pity that my wording was unable to convince even
Quondum. So, I say the same in a less encyclopedic manner. The categorial "zeroness" of {0} actually relies on the fact that pseudo-rings, modules, and vector spaces are
424:. And Quondum suggests the term "zero module" for the zero object. IMHO we should choose the name not more recognizable, but less ambiguous. I am almost sure that the word "trivial" will ultimately be dropped in the title, and
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Just a thought, and I'm sure this will come up in a more general discussion: parenthetical disambiguators are generally discouraged if there is nothing to disambiguate. In this instance, this means we'd have to consider
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This seems to hinge on the implication that a trivial ring is not a true zero object (still beyond my ken). My support for the article's current name was based on the asssumption that these objects all fitted (and that
638:) are also not called zero objects. I'm weak in category theory, so the distinction between its use in algebra and in category theory escapes me. This will determine whether any disambiguator is appropriate. —
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In the category of groups, any trivial group is a zero object. The same is true for the categories of abelian groups, modules over a ring, and vector spaces over a field. This is the origin of the term "zero
378:, which means here to consider only the trivial objects which are used in non trivial math, and to mention the other ones only by sentences like "some authors call this notion in this way". Thus I propose to
1221:, because the reader could believe that the trivial field is a field. I have done a similar edit in this page, but this fundamental information is not enough visible for the public intended by Rumping. I
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article was easily understood by those whose algebra just about extends to groups, rings and fields but does not stretch to further abstraction. I have put it back for the time being --
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I think that "null module" or "zero module" are more usual names for this notion. Moreover, it should be mentioned in the page that this module is frequently used in the context of
973:, and I guess it should behave well as an initial object. So the argument given that ℤ is the initial object in the category of rings does not seem right. I know that
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The fact that the trivial ring is not a zero object in the category of unital rings is not a real objection to call it also a zero ring. Thus it has a natural place in
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Yes you should save your version over it: My edits are easy to restore after that, if they are yet needed (at least may edit of {{tl:Merge}}) will no more be useful).
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But "zero object" is a well established terminology in category theory and something has to be kept for people who looks for information on it. This means a redirect
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with the interpretation of the objection. The ideal would be to get a practitioner in the field to comment on usage of terms. I am definitely not an expert. —
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trivial ring". On the opposite, for groups and similar structures, it depends on the context if the argument of isomorphism is compelling: Every group contains
374:. Note also that the first page considers also a notion of "trivial module structure" which exists only for "non unitary modules". I think that we should apply
485:. But it would make the article difficult to read for non-algebraists, and will shift the topic to category-theoretical stuff from current descriptive stance.
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object. I suggest that the article should refer to "a trivial module", "a trivial vector space" and "a trivial algebra" in recognition of this. Comment? —
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that we are trying to document in this context (abstract algebra). (Forgive my obtuseness – I still need water wings.) I would then support the move of
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distinguished point 0; see my diagram. Inclusion of another constant 1 breaks all the picture, because object are immediately divided to "regular" (
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I was not suggesting any change to that article; it is a very clear article and should not be touched. What I was suggesting is that the redirect
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D.Lazard made a quite inconvenient thing starting to alter the article independently, despite my forecast. Should I save my version over his one?
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I do not get to which exactly realization of the empty matrix the space of possible multiplications is isomorphic. My original version,
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509:. And then, at least, the naming problem in the case of a module is in the corresponding section, not in the name of an article. —
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the corresponding object in each of several categories. It also gives scope for merging the small articles on each, such as
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Exact sequences' 0 may also denote the trivial
Abelian group. But any Abelian group inherently is a ℤ-module, indeed.
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910:. It's evident that this is not intended use (only use one or the other), as one might observe from the following:
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A complete tangle. Proofwiki.org refer as "trivial module" to the thing with zero multiplication, analogous to our
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if we find no conflicting use for the term, for which there is currently a redirect that we would replace. —
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The last line (at least for me) is disproportionately large, whereas the first two lines are the same. —
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Is the trivial algebra also a zero object in some category? I am not an expert in the category theory.
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It should be noted that
Mathworld has two articles on the same object, called either "trivial module"
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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You are correct in wondering whether the merge was the right solution, so I have undone it. The
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I agree to refer to "a trivial module", "a trivial vector space" and "a trivial algebra" and "
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article, but they are not written so clearly and are obscured by more complicated ideas. --
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in ambiguity. What backup names can we use if the problem become severe? I can propose
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the target should be rewritten to be accessible for the public addressed by
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As with most mathematical terms, context seems to be significant. So whereas
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being initial and terminal, and so it is not a direct synonym for the
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by those whose algebra just about extends to groups, rings and fields
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Definition 1.1.1 By a zero object in a category C, we mean an object
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Mention that "trivial module" is also used for this notion
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article requires the reader to understand the concept of
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Mention (and even emphasize) that the zero module is the
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or hatnotes in both pages talking on "zero objects".
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abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ
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abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ
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abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ
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1266:What do I think about it, was expressed in
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376:WP:notability
373:
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353:
351:I've located
350:
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316:singleton set
313:
310:and probably
309:
308:trivial group
305:
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251:trivial space
248:
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179:vector-valued
176:
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96:
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77:
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68:
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61:
56:
52:
46:
38:
34:
29:
28:
19:
1385:
1381:
1365:
1360:
1356:
1354:
1274:
1270:
1234:
1227:trivial ring
1222:
1216:
1213:trivial ring
1211:
1174:trivial ring
1147:
1129:trivial ring
1108:
1104:
1053:Trivial ring
1049:
1010:
998:
995:pointed sets
956:
950:
933:
915:
889:
807:
780:Zero element
745:
741:
731:
729:
683:
622:
602:
574:
572:
549:
507:trivial ring
472:
443:
422:zero algebra
327:
312:trivial ring
240:
221:
198:
178:
174:
172:
148:Low-priority
147:
107:
73:Low‑priority
51:WikiProjects
1297:zero object
1279:Incnis Mrsi
1198:Incnis Mrsi
1152:Incnis Mrsi
1148:improvement
1089:Incnis Mrsi
1015:Incnis Mrsi
999:exactly one
826:Incnis Mrsi
812:Incnis Mrsi
771:Zero object
750:Zero object
742:in terms of
706:zero object
684:zero object
680:zero object
666:Incnis Mrsi
627:zero object
623:zero object
591:Zero object
579:Zero object
551:zero object
535:Incnis Mrsi
487:Incnis Mrsi
430:Incnis Mrsi
395:zero object
383:Zero module
320:zero module
291:Incnis Mrsi
255:Incnis Mrsi
224:Incnis Mrsi
184:Incnis Mrsi
123:Mathematics
114:mathematics
70:Mathematics
41:Start-class
1437:Categories
656:(or, say,
444:Zero space
247:zero space
199:defined by
654:empty set
636:empty set
1420:contribs
1408:asmeurer
1391:D.Lazard
1366:the same
1301:D.Lazard
1239:D.Lazard
949:Section
840:D.Lazard
808:abstract
710:D.Lazard
608:D.Lazard
604:object".
404:D.Lazard
398:module".
274:D.Lazard
1370:Quondum
1321:Quondum
1254:Quondum
1183:Rumping
1133:Rumping
1113:Quondum
1066:history
1030:Quondum
1009:), and
979:Quondum
953:: rings
936:Quondum
877:inside
788:Quondum
688:Quondum
640:Quondum
575:support
556:Quondum
511:Quondum
449:Quondum
356:Quondum
339:Quondum
204:Quondum
150:on the
1233:, but
959:fields
523:I put
335:object
324:module
318:. So
47:scale.
1229:into
1074:watch
1070:links
1007:1 = 0
1003:1 ≠ 0
997:with
786:. —
686:. —
528:Merge
337:. —
1414:talk
1395:talk
1316:diff
1314:The
1305:talk
1283:talk
1243:talk
1202:talk
1187:talk
1156:talk
1137:talk
1093:talk
1078:logs
1062:talk
1058:edit
1019:talk
963:ring
905:math
900:and
895:mvar
882:math
872:mvar
844:talk
830:talk
816:talk
724:here
714:talk
670:talk
612:talk
589:and
539:talk
505:and
491:talk
434:talk
420:and
408:talk
295:talk
278:talk
259:talk
249:and
237:Move
228:talk
188:talk
1382:the
1361:the
1357:the
1225:of
1011:any
967:1≠0
806:to
752:to
629:to
142:Low
1439::
1417:|
1397:)
1307:)
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1021:)
908:}}
902:{{
898:}}
892:{{
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879:{{
875:}}
869:{{
846:)
832:)
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726::
716:)
672:)
614:)
606:—
601::
573:I
541:)
531:}}
525:{{
493:)
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261:)
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1423:)
1411:(
1393:(
1386:a
1326:✎
1303:(
1281:(
1275:1
1271:1
1259:✎
1241:(
1200:(
1185:(
1154:(
1135:(
1118:✎
1091:(
1080:)
1056:(
1035:✎
1017:(
984:✎
941:✎
842:(
828:(
814:(
793:✎
732:0
712:(
693:✎
668:(
645:✎
610:(
561:✎
537:(
516:✎
489:(
454:✎
432:(
406:(
361:✎
344:✎
293:(
276:(
257:(
226:(
209:✎
186:(
154:.
53::
20:)
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