Endomorphism

In .

Given an object X in a category C and two endomorphisms f and g of X , 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 (mathematics), called the endomorphism ring of the object. This is true, for example, in the categories of abelian groups, module (mathematics)s, and vector spaces. In general it is true in all preadditive category.

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

=External links=

*[http://www.mathematics21.org/pseudomorphisms-category.xml Category of Endomorphisms and Pseudomorphisms]. [http://www.mathematics21.org/ Victor Porton]. 2005. - Endomorphisms of a category theory (particularly of a category with partial order category theory) are also category theory of certain categories.