Endomorphism

In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are simply functions from a set S into itself.

In any category, the composition of any two endomorphisms of X is again an endomorphism of X. It follows that the set of all endomorphisms of X forms a monoid, denoted End(X) (or End_C(X) to emphasize the category C).

An invertible endomorphism of X is called an automorphism. The set of all automorphisms is a subset of End(X) with a group structure, called the automorphism group of X and denoted Aut(X). In the following diagram, the arrows denote implication:

Any two endomorphisms of an abelian group A can be added together by the rule (ƒ + g)(a) = ƒ(a) + g(a). Under this addition, the endomorphisms of an abelian group form a ring (the endomorphism ring). For example, the set of endomorphisms of Zn is the ring of all n × n matrices with integer entries. The endomorphisms of a vector space or module also form a ring, as do the endomorphisms of any object in a preadditive category. The endomorphisms of a nonabelian group generate an algebraic structure known as a nearring. Every ring with one is the endomorphism ring of its regular module, and so is a subring of an endomorphism ring of an abelian group, however there are rings which are not the endomorphism ring of any abelian group.