simple module

A simple module is an atom of module theory: a nonzero module with no internal structure to speak of, no submodules except the two it cannot avoid (zero and itself). It is the module-theoretic counterpart of a prime number or a simple group — the indivisible building block out of which more complicated modules are assembled, layer by layer, in a composition series.

An R-module M is simple (or irreducible) if M ≠ 0 and its only submodules are 0 and M. Equivalently, M is generated by any one of its nonzero elements: M = Rm for every nonzero m. By the correspondence between cyclic modules and ideals, simple modules are exactly the modules R/I where I is a maximal left ideal of R. Every nonzero finitely generated module has a simple quotient.

Simple modules are the subject of Schur's lemma: any homomorphism between simple modules is either zero or an isomorphism, so the endomorphism ring of a simple module is a division ring. They are the irreducible representations when R is a group algebra, and the composition factors of any module of finite length. Classifying the simple modules over a ring is, in many settings, the central representation-theoretic problem.