Simple module

Updated 2026-07-30

In mathematics, specifically in ring theory, a simple module over a ring R is a (left or right) module over R that is non-zero and has no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form the building blocks for modules of finite length and are analogous to simple groups in group theory.

Definition

Let R be a ring (assumed to be unital). A (left or right) R-module M is called simple (or irreducible) if:

M ≠ 0 (it is not the zero module), and
The only submodules of M are 0 and M itself.

Examples

Abelian groups (ℤ-modules): The simple ℤ-modules are precisely the cyclic groups of prime order, i.e., ℤ/pℤ for a prime p.
Quotient by a maximal ideal: If I is a right ideal of R, then the quotient module R/I is simple if and only if I is a maximal right ideal. Conversely, every simple R-module is isomorphic to R/m for some maximal right ideal m.
Group representations: If k is a field and G is a group, then the simple modules over the group ring k[G] are precisely the irreducible representations of G.

Basic Properties

Simple modules are precisely the modules of length 1.
Every simple module is indecomposable, but the converse is not true in general.
Every simple module is cyclic (generated by a single element).
Not every module has a simple submodule; for example, the ℤ-module ℤ has no simple submodule.

Schur's Lemma

A fundamental result about simple modules is Schur's lemma: If M and N are simple modules over the same ring, then any module homomorphism f: M → N is either the zero map or an isomorphism. Consequently, the endomorphism ring of any simple module is a division ring.

Simple Modules and Composition Series

A composition series of a module M is a finite chain of submodules

0 = M₀ ⊂ M₁ ⊂ ⋯ ⊂ Mₙ = M

such that each quotient Mᵢ/Mᵢ₋₁ is simple. The Jordan–Hölder theorem guarantees that any two composition series of a module have the same length and, up to permutation and isomorphism, the same composition factors. Simple modules thus serve as the irreducible "atoms" from which finite-length modules are built.

The Jacobson Density Theorem

The Jacobson density theorem states that if U is a simple right R-module and D = EndR(U) is its endomorphism division ring, then for any D-linear operator A on U and any finite D-linearly independent subset X of U, there exists an element r ∈ R such that x·A = x·r for all x ∈ X. A corollary is Wedderburn's theorem: any right Artinian simple ring is isomorphic to a full matrix ring over a division ring.