Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map
[1][2] 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.
In a similar fashion, if R is any commutative ring, the endomorphisms of an R-module form an algebra over R by the same axioms and derivation.
With the additional operation of composition of homomorphisms, End(A) is a ring with multiplicative identity.
[3] However, the closure of the set of endomorphisms under the above operations is a canonical example of a near-ring that is not a ring.