How do I give a good sketch of the Artin-Wedderburn Theorem?

Question

I have an oral exam coming up, and it's definitely possible A-W comes as a question due to its importance. However, the proof of the theorem is rather long, it has many claims, it's pretty technical, etc. It's a difficult proof, and I have no clue how to give a good sketch of it. What should I keep in mind? The only lemma we use is if $A$ is a minimal left ideal in a ring $R$ then $A^2 = 0$ or $A = Re, e$ an idempotent in $R$

Answer 1

Try this link that includes a short (less than two pages of pure proof content) and well structured proof based on Weddeburn's Theorem for simple ring with a minimal left ideal (with a short proof by Henderson based on Brauer's Lemma) and by introducing a weak finiteness condition in a ring (equivalent to the ring containing no infinite orthogonal set of idempotents) to obtain the extension to semiprime left Artinian rings.

Answer 2

From a representation-theoretic point of view, Artin-Wedderburn is pretty straightforward. My preference is for right modules, writing homomorphisms on the left. Then for every ring $R$ one has $R\cong\mathrm{End}_R(R)$. Next, if $S$ is a simple $R$-module, then $\mathrm{End}_R(S)$ is a division ring, and there are no homomorphisms between non-isomorphic simples. Thus if $$ M = \bigoplus_{i=1}^n S_i^{d_i} $$ is a semisimple $R$-module, with $S_i$ pairwise non-isomorphic simples, then $$ \mathrm{End}_R(M) \cong \prod_{i=1}^n \mathbb M_{d_i}(\mathrm{End}_R(S_i)) $$ is a product of matrix rings over division rings. Moreover, one sees that there is one factor for each simple, and the size $d_i$ equals the multiplicity.

Thus one gets in a fairly elementary way the result that if the regular module $R_R$ is semisimple, then $R$ is isomorphic to a finite product of matrix rings over division rings.

Of course, one often also wants the ring-theoretic characterization of when the regular module is semisimple, which is when the ring is (right) artinian with vanishing Jacobson radical.