Home > Endomorphism

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 categoriesA preadditive category is a category that is enriched over the monoidal category of abelian groups. In other words, the category C is preadditive if every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinea.

An endomorphism that is also an isomorphismIn mathematics, an isomorphism is a kind of interesting mapping between objects. Douglas Hofstadter provides an informal definition: :The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part is termed an automorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an obj. In the following diagram, the arrows arrows denote implication.