Simple module

In mathematics, more specifically modern algebra and module theory, a (left or right) module U over a ring R is a simple module, if there are no non-trivial proper submodules of U (over R). Equivalently, U is a simple module over R if and only if the cyclic submodule generated by every non-zero element on U equals U.

Simple modules, in some sense, form the "building blocks" for modules of finite length, analogous to the fact that finite simple groups form the building blocks for all finite groups. With this perspective, the understanding of simple modules is readily seen to be an important aspect of module theory.

An important advance in the theory of simple modules was the Jacobson density theorem. The Jacobson density theorem states: Let U be a simple right R-module and write D = End_R(U). Let A be any D-linear operator on U and let X be a finite D-linearly independent subset of U. Then there exists an element r of R such that x·A = x·r for all x in X.. In particular, any primitive ring may be viewed as (that is, isomorphic to) a ring of D-linear operators on some D-space.

A consequence of the Jacobson density theorem, is Wedderburn's theorem; namely that any right artinian simple ring is isomorphic to the matrix ring over a full division ring. This can also be established as a corollary of the Artin–Wedderburn theorem.

In this article, all modules are assumed right unital modules over some ring R (as opposed to "left module over R").

Examples

Abelian groups are the same as Z-modules. The simple Z-modules are precisely the cyclic groups of prime order.

If K is a field and G is a group, then a group representation of G is a left module over the group ring KG. The simple KG modules are also known as irreducible representations. A major aim of representation theory is to list those irreducible representations for a given group.

Given a ring R and a left ideal I in R then I is a simple R-module if and only if I is a minimal left ideal in R (does not contain any other non trivial left ideals). The factor module R/I is a simple R-module if and only if I is a maximal left ideal in R (is not contained in any other non-trivial left ideals).

Properties

The simple modules are precisely the modules of length 1; this is a reformulation of the definition.

Every simple module is indecomposable, but the converse is in general not true.

Every simple module is cyclic, that is it is generated by one element

Not every module has a simple submodule; consider for instance the Z-module Z in light of the first example above.

Let M and N be (left or right) modules over the same ring, and let f : M → N be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective because the kernel of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image of f is a submodule of N. If M = N, then f is an endomorphism of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism. Consequently the endomorphism ring of any simple module is a division ring. This result is known as Schur's lemma.

The converse of Schur's lemma is not true in general. For example, the Z-module Q is not simple, but its endomorphism ring is isomorphic to the field Q.

See also

• Semisimple modules are modules that can be written as a sum of simple submodules
• Simple groups are similarly defined to simple modules
• Irreducible ideal.