Module endomorphism (${\mathrm{End}}_R(M)$)

A module endomorphism is a module homomorphism from a module to itself. These maps form the endomorphism ring ${\mathrm{End}}_R(M)$ under pointwise addition and composition. An invertible module endomorphism is a module automorphism.

A module automorphism is an invertible module endomorphism. Equivalently, it is a bijective module homomorphism from a module to itself; its inverse is automatically a module homomorphism. They form a group under composition.