A classification theorem stating that any semisimple Artinian ring (equivalently any finite-dimensional semisimple algebra over a field) is isomorphic to a finite direct product of matrix algebras over division rings; in particular, simple artinian rings are matrix algebras over division rings.

Principle

The organizing idea is that semisimplicity forces a decomposition into simple components, and each simple component has the form End_D(V) ≅ M_n(D) for a division algebra D and finite-dimensional vector space V over D; central idempotents separate factors.

Demonstration

Illustration: the group algebra C[G] of a finite group over the complex numbers decomposes (by Maschke) as a finite product of full matrix algebras over C, one matrix algebra for each irreducible complex representation, giving the explicit block decomposition used in representation theory.

Misapplication

Applying the theorem to non-artinian or non-semisimple rings (for example infinite-dimensional algebras with nonzero Jacobson radical) and expecting a matrix-product decomposition; this ignores radicals, topological completions, or infinite direct product subtleties.

Consequence

The theorem yields a complete structural description of semisimple algebras: classification of simple modules, identification of centers via block decomposition, explicit Morita-type equivalences, and reduction of representation questions to linear algebra over division rings.

Reversal

If a ring is not semisimple or not Artinian, the Artin–Wedderburn conclusion fails: there may be a nontrivial Jacobson radical, indecomposable modules beyond matrix blocks, and no finite product decomposition into matrix algebras.

Boundary

Applies to semisimple Artinian rings and finite-dimensional semisimple algebras over fields; it does not apply to infinite-dimensional rings in general, to rings with nonzero radical, or to algebras over nonfields without finiteness conditions unless appropriately modified.

Semantic Tension

There is tension between 'simple' and 'semisimple' usages and between finite-dimensional algebraic statements and infinite or topological analogues; another nearby confusion is with Wedderburn's little theorem about finite division rings being fields, which is a distinct statement.

Synthesis

Artin–Wedderburn identifies semisimple Artinian rings with finite products of matrix algebras over division rings, reducing structural and representation-theoretic questions to linear algebra over those division algebras and clarifying how central idempotents partition the algebra into blocks.