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:

automorphism | isomorphism | |

endomorphism | (homo)morphism |

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 **Z***n* 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.

Read more about Endomorphism: Operator Theory, Endofunctions in Mathematics

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