Definition:Ring of Endomorphisms

Theorem
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 a ring with unity, called the ring of endomorphisms of the abelian group $\struct {G, \oplus}$.

Proof
By Structure Induced by Group Operation is Group, $\struct {\mathbb G, \oplus}$ is an abelian group.

By Set of Homomorphisms is Subgroup of All Mappings, it follows that $\struct {\mathbb G, \oplus}$ is a subgroup of $\struct {G^G, \oplus}$.

Next, we establish that $*$ is associative.

By definition, $\forall u, v \in \mathbb G: u * v = u \circ v$ where $u \circ v$ is defined as composition of mappings.

Associativity of $*$ follows directly from Composition of Mappings is Associative.

Next, we establish that $*$ is distributive over $\oplus$.

Let $u, v, w \in \mathbb G$.

Then:
 * $\paren {u \oplus v} * w = \paren {u \oplus v} \circ w$
 * $u * \paren {v \oplus w} = u \circ \paren {v \oplus w}$

So let $x \in G$.

Then:

So $\paren {u \oplus v} * w = \paren {u * w} \oplus \paren {v * w}$.

Similarly:

So:
 * $u * \paren {v \oplus w} = \paren {u * v} \oplus \paren {u * w}$

Thus $*$ is distributive over $\oplus$.

The ring axioms are satisfied, and $\struct {\mathbb G, \oplus, *}$ is a ring.

The zero is easily checked to be the mapping which takes everything to the identity:


 * $e: G \to \set {e_G}: \map e x = e_G$

The unity is easily checked to be the identity mapping, which is known to be an automorphism.