Talk:Artin–Wedderburn theorem

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Explanation of removing "rings satisfying these equivalent definitions are artinian semisimple"[edit]

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 semisimple rings. Moreover, the use of "equivalent descriptions" was also unclear in the deleted sentence... I could not see any more than one description. Rschwieb (talk) 12:01, 13 October 2012 (UTC)

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):

  • Kelarev, p. 13: "A ring R is said to be semisimple or ρ-semisimple 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 semiprimitive or semiprime ring, respectively. Semiprimitive rings are also often called Jacobson semisimple (or semisimple) rings. A ring is right Artinian (Noetherian) 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 semisimple Artinian ring."
  • 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) RR is a semisimple module; (3) R is isomorphic to a finite sum of rings of n × n matrices over division rings. [Proof ...] Defintion 3.3.3: A ring which satisfies the conditions of Theorem 3.3.2 is said to be semisimple Artinian. [...] 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; [...]"
  • Anderson-Fuller, 2nd ed., p. 153: "A ring R is said to be semisimple in case the left regular module RR 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 RR is semisimple" iff "R is simple and RR is semisimple".
  • McConnel and Robson, rev. ed., p.3 "1.9 The next few facts connect module properties with ring properties. They comprise the Artin–Wedderburn theory. [Schur's lemma] 1.10 [definition of simple ring, then:] Theorem. The following conditions on R are equivalent: (i) R is a simple right Aritnian ring; (ii) R ≃ Mn(D) for some n and some (uniquely determined) division ring D; [(iii) omitted here]. Note that (ii) is symmetric. So the left-hand version of (i) and (iii) are also valid, and hence R is Artinian. 1.11 [defines nilpotent ideal, then:] Theorem. The following conditions on R are equivalent (i) R is a finite direct product of simple Artinian rings; (ii) RR is semisimple; (iii) Every right R-module is semisimple; [two more, omitted]. Note again the symmetry, provided here by (i) and 1.10. Such a ring R is called a semisimple (Artinian) ring."

-- Tijfo098 (talk) 14:04, 13 October 2012 (UTC)

Outside this article, we have semisimple algebra, semisimple ring, semisimple Artinian, Jacobson semisimple ring and also Jacobson ring, 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. D.Lazard (talk) 19:03, 13 October 2012 (UTC)
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 conditions on an X are equivalent to it being P: B,C,D". Deltahedron (talk) 20:06, 13 October 2012 (UTC)
@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 define 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 define a semisimple ring as a finite product of matrix rings over division rings. Every source I have defines them in a different way, and then proves this theorem about their structure.
@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 it makes more sense to make the theorem page the defining one. I think it makes a lot more sense to let the semisimple ring/ algebra pages discuss the definition.
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. Rschwieb (talk) 13:11, 15 October 2012 (UTC)