Semisimple rings and Artin-Wedderburn

What semisimple means

A ring R is semisimple if, viewed as a left module over itself, it is a direct sum of simple modules. Equivalently — and this is the workable definition — every left R-module is completely reducible: it splits as a direct sum of simple modules, so every submodule is a direct summand. There are no extensions to untangle, no nontrivial short exact sequences that fail to split. Life is, locally, as easy as linear algebra over a field.

Schur's lemma as the engine

The whole structure theory runs on one tiny observation. Schur's lemma: a homomorphism between simple modules is either zero or an isomorphism. Therefore the endomorphism ring End_R(S) of a simple module S is a division ring — every nonzero endomorphism is invertible. Division rings appear not by decree but as endomorphism rings of simple modules. That is where the D in M_n(D) is going to come from.

Combine Schur with one more idea: for any R-module M, End_R(M^n) ≅ M_n(End_R(M)). Endomorphisms of a direct sum of n copies are organized into an n×n grid of component maps, exactly like a matrix. Put these two facts side by side and the shape M_n(D) is already inevitable.

The Artin-Wedderburn theorem

Here is the climax. The Artin-Wedderburn theorem: a semisimple ring R is isomorphic to a finite direct product of matrix rings over division rings, R ≅ M_{n_1}(D_1) × ⋯ × M_{n_r}(D_r). The number r equals the number of isomorphism classes of simple modules; the D_i are their endomorphism division rings (up to op); and the n_i and D_i are uniquely determined. The simple semisimple rings are exactly the single factors M_n(D).

1. Decompose R = ⊕ L_i as a left module into simple left ideals, and group the L_i into isotypic blocks — those isomorphic to the same simple module S_j.
2. Each isotypic block is a two-sided ideal B_j, and R = B_1 × ⋯ × B_r as a ring (the blocks annihilate one another).
3. Compute R = End_R(R) using the dual to the module action; the block B_j becomes End_R(S_j^{n_j}) ≅ M_{n_j}(D_j) with D_j = End_R(S_j) by Schur.
4. Uniqueness: the simple modules and their multiplicities are invariants of R, so (n_j, D_j) are determined up to ordering and op.

Example: take complex group algebra C[S_3], dimension 6.
  S_3 has three irreducible reps over C:
    trivial   (dim 1)
    sign      (dim 1)
    standard  (dim 2)
Artin-Wedderburn forces
  C[S_3]  ≅  M_1(C) × M_1(C) × M_2(C).
Dimension check:  1^2 + 1^2 + 2^2 = 1 + 1 + 4 = 6.  Matches |G| = 6.

The rule  sum of (dim of irreducible)^2 = |G|  is exactly the
Wedderburn dimension count for the semisimple ring C[G].