# Wedderburn-Artin theorem

There is a website with a short proof of Wedderburn-Artin theorem link. I am not sure if the proof is ok and in fact I haven't understood what the following lemma means:

Lemma 1. If $M$ is a cyclic left $R$-module satisfying the descending chain condition on submodules and generated by a set of isomorphic copies of a single $R$-module $S$, then $M$ is isomorphic to a direct sum of nitely many copies of S .

Could someone clarify me what means "cyclic and generated by a set of isomorphic copies of a single $R$-module $S$ ? Otherwise, what is a good reference for a proof of this theorem?

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A cyclic left $R$-module is an $R$-module of the form $Ra = \{ r \cdot a \}$ with $a$ being some element of the module (think of the similar idea with cyclic groups, generated by only one element ; here you only need the module operations and that element $a$ to generate $Ra$, where as in group theory you only need the element $x$ generating $\langle r \rangle$ using the group operation and this element).

By "generated by a set of isomorphic copies of a single $R$-module $S$", the author means your $R$-module $A$ is such that there is a family of subsets of your $R$-module, call it $C$, such that when you take the $R$-module that they span, you get your original $R$-module $A$, but all the subsets in $C$ are isomorphic to that module $S$. For instance, a $R$-module $A$ generated by (finitely many) isomorphic copies of $Ra$ would be such that there exists $a_1, \dots, a_n \in A$ such that $$A \cong Ra_1 + R a_2 + \dots + Ra_n = \left\{ \sum_{i=1}^n r_i a_i \, | \, r_i \in R \right\}$$ and $Ra_i \cong Ra$ (which doesn't mean $a_i = a$).

There are many proofs of the Wedderburn-Artin theorem, but depending on your level of mathematics or your degree of understanding abstraction you can find proofs that are more suitable if you want to take it easy or more appropriate proofs to find the most general statements. You need to "shop" a little bit. I didn't get into that for a while so I'm sorry if I can't help about this part, but this was definitely too long for a comment.

Hope that helps,

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The module $B$ is a typo, right? You mean $A$ I guess. – Binai Nov 20 '12 at 2:47
Patrick: Thanks. I have some good level in mathematics and I have never heard this second part which I mentioned. The cyclic part is of course OK. But thanks in clarify me the second one! – Binai Nov 20 '12 at 2:48
@Matt : Yes, typo. I changed notation along the way but missed that one. For a general proof of Wedderburn-Artin, I suggest you read the Dummit & Foote. It might require some work to understand it fully though ; the results they show depend on what they have proved beforehand ; if you are not familiar with the usual statements it might take you some time to understand everything that happens. It's mostly harder than usual to get it because the steps of the proof are left as exercises, but there are big hints to get there. – Patrick Da Silva Nov 20 '12 at 6:24

"generated by a set of isomorphic copies of a single $R$-module $S$" means that there exists a family $(S_i)_{i\in I}$ of submodules of $M$ such that $M=\sum_{i\in I}S_i$ and $S_i\cong S$ for all $i\in I$. (Btw, beside this Lady's proof is nice and short.)

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