Definition:Ring of Endomorphisms

Definition
Let $\struct {G, \oplus}$ be an abelian group.

Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$.

Let $*: \mathbb G \times \mathbb G \to \mathbb G$ be the operation defined as:
 * $\forall u, v \in \mathbb G: u * v = u \circ v$

where $u \circ v$ is defined as composition of mappings.

Then $\struct {\mathbb G, \oplus, *}$ is called the ring of endomorphisms of the abelian group $\struct {G, \oplus}$.

Also see

 * Ring of Endomorphisms is Ring with Unity
 * Ring of Endomorphisms is not necessarily Commutative Ring
 * Set of Endomorphisms of Non-Abelian Group is not Ring