Endomorphisms

An endomorphism is a structure-preserving map from a mathematical object to itself. If X carries an algebraic structure such as a group, ring, module, or vector space, an endomorphism is a homomorphism f: X → X that preserves the relevant operations. The set of all endomorphisms of X is usually denoted End(X).

Examples illustrate the concept. For the additive group of integers (Z, +), every endomorphism is given by multiplication by a fixed integer n, so End(Z) is isomorphic to the ring of integers Z. For a finite cyclic group, endomorphisms correspond to multiplication by a fixed integer modulo the group order. In linear algebra, End(V) for a vector space V over a field F is the set of all linear maps V → V. If V has dimension n, End(V) is isomorphic to the matrix ring M_n(F). End(V) forms a ring under pointwise addition and composition; bijective endomorphisms are automorphisms, and Aut(V) = GL(V).

More generally, if M is an R-module, End_R(M) denotes the endomorphisms of M as an R-module. End_R(M) is a ring under addition and composition, and it encodes many structural features of M.

Key properties include kernels and images. For a group endomorphism f, ker f is a normal subgroup; for a vector space endomorphism, ker f is a subspace and rank-nullity gives dim im f + dim ker f = dim V. Endomorphisms thus provide a framework for studying internal structure, invariants, and decompositions of algebraic objects.