Endomorphism ring

Summary

The endomorphism ring of a module is the set of all module homomorphisms from the module to itself, with pointwise addition and composition as multiplication. It is the main source of noncommutative rings: every ring is a subring of the endomorphism ring of its own additive group. Properties of a module are often read off its endomorphism ring. For example, a nonzero module is indecomposable exactly when its endomorphism ring has no idempotents other than $0$ and $1$. For a vector space it is the algebra of endomorphisms, and for $K^n$ it is the matrix ring $M_n(K)$.