Definition:Endomorphism Ring/Abelian Group

Definition

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

Let $\map {\mathrm {End} } G$ be the set of endomorphisms of $G$.

The endomorphism ring of $G$ is the algebraic structure:

$\struct {\map {\mathrm {End} } G, \oplus, \circ}$

where:

$\circ$ denotes composition of mappings

$\oplus$ denotes the pointwise operation on $\map {\mathrm {End} } G$ induced by $+$.

Also denoted as

When presenting the endomorphism ring of an abelian group $\struct {G, +}$, it is commonplace to use the same symbol for the pointwise operation on $\map {\mathrm {End} } G$ as for the binary operation which induced it.

Hence it would be presented as:

$\struct {\map {\mathrm {End} } G, +, \circ}$

Both forms may be encountered on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see

• Endomorphism Ring of Abelian Group is Ring with Unity: $\struct {\map {\mathrm {End} } G, \oplus, \circ}$ is shown to be a ring.

• Results about endomorphism rings of abelian groups can be found here.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts