Endomorphism

In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. So, for example, an endomorphism of a vector space V is a linear map f : V → V and an endomorphism of a group G is a group homomorphism f : G → G, etc. In general, we can talk about endomorphisms in any category.

Given an object X in a category C and two endomorphisms f and g of X, then the functional composition f O g is also an endomorphism of X. Since the identity map on X is also an endomorphism of X, we see that the set of all endomorphisms of X forms a monoid, denoted End_C(X) or just End(X) if the category is understood.

In many but not all situations it is possible to add endomorphisms, and the endomorphisms of a given object then form a ring, called the endomorphism ring of the object. This is true, for example, in the categories of abelian groups, modules, and vector spaces. In general it is true in all preadditive categories.

An endomorphism that is also an isomorphism is termed an automorphism. In the following diagram, the arrows arrows denote implication.

automorphism ----> isomorphism
     |                  |
     |                  |
     V                  V
endomorphism ----> (homo)morphism