Endomorphism ring

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

An abelian group is the same thing as a module over the ring of integers, which is the initial ring. In a similar fashion, if R is any commutative ring, the endomorphism monoids of its modules form algebras over R by the same axioms and derivation. In particular, if R is a field F, its modules M are vector spaces V and their endomorphism rings are algebras over the field F.

If the set A does not form an abelian group, then the above construction is not necessarily additive, as then the sum of two homomorphisms need not be a homomorphism. This set of endomorphisms is a canonical example of a near-ring that is not a ring.