Editing Endomorphism (section)

==Endomorphism rings==
{{main|Endomorphism ring}}
Any two endomorphisms of an [[abelian group]], {{math|''A''}}, can be added together by the rule {{math|(''f'' + ''g'')(''a'') {{=}} ''f''(''a'') + ''g''(''a'')}}.  Under this addition, and with multiplication being defined as function composition, the endomorphisms of an abelian group form a [[ring (mathematics)|ring]] (the [[endomorphism ring]]).  For example, the set of endomorphisms of <math>\mathbb{Z}^n</math> is the ring of all {{math|''n'' × ''n''}} [[Matrix (mathematics)|matrices]] with [[integer]] entries.  The endomorphisms of a vector space or [[module (mathematics)|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;<ref>Jacobson (2009), p. 162, Theorem 3.2.</ref> however there are rings that are not the endomorphism ring of any abelian group.