I is a maximal right ideal. Therefore, M is isomorphic to a quotient of R by a maximal right ideal. If k is a field and G is a group, then a group representation of G is a left module over the group ring k[G] (for details, see the main page on this relationship). The simple k[G]-modules are also known as irreducible representations. A major aim of representation theory is to understand the irreducible representations of groups. Basic properties of simple modules 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 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. Simple modules and composition series If M is a module which has a non-zero proper submodule N, then there is a short exact sequence A common approach to proving a fact about M is to show that the fact is true for the center term of a short exact sequence when it is true for the left and right terms, then to prove the fact for N and M/N. If N has a non-zero proper submodule, then this process can be repeated. This produces a chain of submodules In order to prove the fact this way, one needs conditions on this sequence and on the modules Mi /Mi +&hairsp;1. One particularly