312:, with very few links between them. This article is not confusing, because of the footnote explaining "(artinian)". IMO, the terminology is not fixed in the literature, but, because of the importance of W.A. theorem, the use of "semisimple" for "semisimple Artinian" is dominant (that is that I was teached during my studies, where only "semisimple ring" and "Jacobson ring" were used; but it was in France and Bourbaki's terminology was the standard). But, whichever terminology we use, it should be said that "some authors use ..." where needed. It appears also important to reduce the number of articles devoted to only two strongly related concepts.
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462:. More precisely, the introduction of the beginning of Chapter 1 has described briefly the history of the Wedderburn-Artin Theorem. According to the introduction, I believe that the reason why it is more common to say "Wedderburn-Artin Theorem" is that the theory was first developed by Wedderburn and then generalized by Artin. Moreover, almost all researchers in mathematical community call this theorem as "Wedderburn-Artin Theorem". We can search some of them on Zbmath here
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263:(D) for some n and some (uniquely determined) division ring D; . Note that (ii) is symmetric. So the left-hand version of (i) and (iii) are also valid, and hence R is Artinian. 1.11 Theorem. The following conditions on R are equivalent (i) R is a finite direct product of simple Artinian rings; (ii) R
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I took this out because it is circular to say "The Artin
Wedderburn theorem determines the structure of semisimple artinian rings as products of matrix rings. A ring which satisfies these (equivalent) descriptions is sometimes called semisimple Artinian". The A-W Theorem is not used to define
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It is not unknown for texts to write something like "Theorem: The following are equivalent properties of an X: A,B,C,D. We define an X to be P if it satisfies the equivalent conditions on X". It is perhaps more common to see "We define an X to be P if it satisfies A. Theorem: the following
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This doesn't seem to solve the ambiguity of what "semisimple ring" means. It only makes the page more confusing, in my opinion. And the defintion of "semisimple
Artinian" by the equivalent properties of the theorem is not circular for some authors (including the ones you deleted):
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I tried to improve the footnote because I found it a bit confusing. I wanted to emphasize that with
Knowledge's definition of "semisimple", it automatically follows that every semisimple ring is Artinian, though some authors use the term differently.
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R is semisimple. By (13.5) we have that every simple artinian ring is semisimple." Prop 13.5 is a long list of equivalent properties, including that "R is simple and left
Artinian" iff "R is simple and right Artinian" iff "R is simple and
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I added a sketch of the proof of the
Wedderburn-Artin theorem, and moved the links to two other proofs that appear elsewhere on Knowledge to the end of that proof. By the way, there was a completely bogus proof of Wedderburn-Artin on
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semisimple rings. I didn't think it was appropriate also because there was no more than one description mentioned (being a product of matrix rings over division rings). Secondly, I do not believe that it is at all standard to
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I would be very willing to help make the clarification about semisimple rings more clear. I can see how my last version might be improved in some ways. Feedback is welcome. (I'll start with the feedback given above, too.
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R is a semisimple module; (3) R is isomorphic to a finite sum of rings of n × n matrices over division rings. Defintion 3.3.3: A ring which satisfies the conditions of
Theorem 3.3.2 is said to be
365:@Tijfo098 I never meant to dispute that it is an equivalent condition to being semisimple. What I'm disputing is that nobody uses it as a definition. I also mean to dispute your edit summary that
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217:) if every descending (ascending) chain of right ideals in the ring stabilizes. A semiprimitive ring is right Artinian if and only if it is left Artinian, and so it is called a
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This is a famous theorem which was due to two mathematicians
Wedderburn and Artin. In the literature, it is more common to refer this theorem as "Wedderburn-Artin theorem".
349:@Deltahedron While I'm perfectly familiar with that phenomenon, and that is exactly what the Artin-Wedderburn Theorem is, I wanted to say that it shouldn't be used to
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semisimple rings. Moreover, the use of "equivalent descriptions" was also unclear in the deleted sentence... I could not see any more than one description.
232:. Corollary 3.3.4. The following conditions are equivalent for a ring with identity: (1) R is semisimple Artinian; (2) R is left Artinian and semiprime; "
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if ρ(R) = 0. For every radical ρ, the quotient ring R/ρ(R) is semisimple. If ρ is the
Jacobson or the Baer radical, then a ρ-semisimple ring is called a
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Beachy, p. 156: "Theorem 3.3.2 (Artin–Wedderburn): For any ring R the following conditions are equivalent: (1) R is left
Artinian and J(R)=(0); (2)
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is semisimple; (iii) Every right R-module is semisimple; . Note again the symmetry, provided here by (i) and 1.10. Such a ring R is called a
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605:, which many editors had been complaining about since 2005. I deleted that proof and added more information about simple rings.
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Sources are needed, such as number of hits returned by
Scholar Google. in any case "Wedderburn-Artin theorem" redirects here.
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McConnel and Robson, rev. ed., p.3 "1.9 The next few facts connect module properties with ring properties. They comprise the
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ring, respectively. Semiprimitive rings are also often called Jacobson semisimple (or semisimple) rings. A ring is
259:. 1.10 Theorem. The following conditions on R are equivalent: (i) R is a simple right Aritnian ring; (ii) R ≃ M
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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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a semisimple ring as a finite product of matrix rings over division rings. Every source I have
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Explanation of removing "rings satisfying these equivalent definitions are artinian semisimple"
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I suggest to read the book "A first course in non-commutative rings" which can be found here
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seem the OP has a point: W-A is more common. For example, encyclopedia of math uses W-A
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If there is no objection in a few days, I will be implementing the renaming. --
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It seems that there is no way to change the title of an article in Knowledge.
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them in a different way, and then proves this theorem about their structure.
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The title of the article should be "Wedderburn-Artin theorem"
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it makes more sense to make the theorem page the defining one
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conditions on an X are equivalent to it being P: B,C,D".
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Anderson-Fuller, 2nd ed., p. 153: "A ring R is said to be
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https://www.zbmath.org/?q=wedderburn-artin+theorem+
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485:. I support the renaming. —-
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