Endomorphisms
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, f: V → V, and an endomorphism of a group, G, is a group homomorphism f: G → G. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to 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 EndC(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 (f + g)(a) = f(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 near-ring. 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 that are not the endomorphism ring of any abelian group.
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