In this article, all modules will be assumed to be right unital modules over a ring R. Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups.
Conversely, if I is not minimal, then there is a non-zero right ideal J properly contained in I. J is a right submodule of I, so I is not simple.
Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x.
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 + 1.
One particularly useful condition is that the length of the sequence is finite and each quotient module Mi /Mi + 1 is simple.
In this case the sequence is called a composition series for M. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module.
For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull–Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.
An important advance in the theory of simple modules was the Jacobson density theorem.