Editing Endomorphism ring (section)

{{Short description|Endomorphism algebra of an abelian group}}
In [[mathematics]], the [[endomorphism]]s of an [[abelian group]] ''X'' form a [[ring (mathematics)|ring]]. This ring is called the '''endomorphism ring''' of ''X'', denoted by End(''X''); the set of all [[homomorphism]]s of ''X'' into itself. Addition of endomorphisms arises naturally in a [[Pointwise#Pointwise operations|pointwise]] manner and multiplication via [[function composition|endomorphism composition]].  Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the [[zero morphism|zero map]] <math display="inline">0: x \mapsto 0</math>  as [[additive identity]] and the [[identity map]] <math display="inline">1: x \mapsto x</math> as [[Identity element|multiplicative identity]].{{sfn|ps=none|Fraleigh|1976|p=211}}{{sfn|ps=none|Passman|1991|pp=4–5}}

The functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the [[category (mathematics)|category]] of the object under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the endomorphism ring is often an [[algebra (ring theory)|algebra]] over some ring ''R,'' this may also be called the '''endomorphism algebra'''.

An abelian group is the same thing as a [[Module (mathematics)|module]] over the ring of [[integer]]s, which is the [[initial object]] in the [[category of rings]].  In a similar fashion, if ''R'' is any [[commutative ring]], the endomorphisms of an ''R''-module form an [[algebra over a ring|algebra over ''R'']] by the same axioms and derivation.  In particular, if ''R'' is a [[Field (mathematics)|field]], its modules ''M'' are [[Vector space|vector spaces]] and the endomorphism ring of each is an [[Algebra over a field|algebra over the field]] ''R''.