Endomorphism ring of an abelian group

Definition
Suppose $$A$$ is an abelian group. The endomorphism ring of $$A$$, denoted $$\operatorname{End}(A)$$ is defined as follows:


 * 1) As a set, it is the set of all defining ingredient::endomorphisms of $$A$$.
 * 2) The addition is pointwise addition in the target group. In other words, for endomorphisms $$f,g$$ of $$A$$, we define $$f + g$$ as the map $$a \mapsto f(a) + g(a)$$. Thus, the additive identity is the zero map (the map sending everything to zero) and the negation is the pointwise negation.
 * 3) The multiplication is given by function composition. In other words, $$fg$$ is the map sending $$a$$ to $$f(g(a))$$. The identity for multiplication is the identity map.

Facts

 * For a non-abelian group, the pointwise group multiplication of two endomorphisms need not be an endomorphism.
 * For functions that are not endomorphisms, only one of the distributivity laws holds in general.