Schur's Lemma - Generalization To Non-simple Modules

Generalization To Non-simple Modules

The one module version of Schur's lemma admits generalizations involving modules M that are not necessarily simple. They express relations between the module-theoretic properties of M and the properties of the endomorphism ring of M.

A module is said to be strongly indecomposable if its endomorphism ring is a local ring. For the important class of modules of finite length, the following properties are equivalent (Lam 2001, §19):

• A module M is indecomposable;
• M is strongly indecomposable;
• Every endomorphism of M is either nilpotent or invertible.

In general, Schur's lemma cannot be reversed: there exist modules that are not simple, yet their endomorphism algebra is a division ring. Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. However, even over the ring of integers, the module of rational numbers has an endomorphism ring that is a division ring, specifically the field of rational numbers. Even for group rings, there are examples when the characteristic of the field divides the order of the group: the Jacobson radical of the projective cover of the one-dimensional representation of the alternating group on five points over the field with three elements has the field with three elements as its endomorphism ring.