semisimple ring

Some rings shatter cleanly into independent pieces with no leftover debris; others have a sticky, glued-on part that resists decomposition. Semisimple rings are the clean ones. Every module over them falls apart completely into simple pieces, with nothing stuck together — there is no internal radical, no hidden nilpotent gunk obstructing the decomposition. They are the rings where representation theory works as nicely as possible.

Precisely, a ring R is semisimple if R, viewed as a left module over itself, is a direct sum of simple modules. Several equivalent conditions make this powerful: R is semisimple if and only if every left R-module is a direct sum of simple modules, if and only if every short exact sequence of R-modules splits, if and only if every R-module is projective. The condition is left-right symmetric, so the word needs no side.

Structurally, the Artin-Wedderburn theorem pins these down completely: a ring is semisimple if and only if it is a finite direct product of matrix rings M_{n_i}(D_i) over division rings D_i. So a semisimple ring is exactly a finite list of full matrix algebras glued by direct product, and its modules are completely understood. Semisimple rings are automatically Artinian and Noetherian.

The bridge to representation theory is Maschke's theorem: the group algebra k[G] of a finite group G over a field k is semisimple precisely when the characteristic of k does not divide the order of G. This is why complex representations of finite groups decompose into irreducibles, and why modular representation theory in dividing characteristic is harder — there the radical reappears.