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

• Results about Rings of Endomorphisms can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains: Example $21.2$
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $10$