Simple module

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In abstract algebra, a (left or right) module S over a ring R is called simple or irreducible if it is not the zero module 0 and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because these modules form the "building blocks" of all other modules in a certain sense.

[edit] 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).

[edit] 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.

If S is a simple module and f : S → T is a module homomorphism, then f is either the zero homomorphism or injective. This is because the kernel of f is a submodule of S and thus is, by the definition of a simple module, either 0 or S. If T is also a simple module, then f is either zero or an isomorphism. This is because the image of f is a submodule of T and thus is either 0 or T. Taken together, this implies that 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.