Definition:Ring of Endomorphisms

Theorem
Let $\left({G, \oplus}\right)$ be an abelian group.

Let $\mathbb G$ be the set of all group endomorphisms of $\left({G, \oplus}\right)$.

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 $\left({\mathbb G, \oplus, *}\right)$ is a ring with unity, called the ring of endomorphisms of the abelian group $\left({G, \oplus}\right)$.

Proof

 * By Induced Group, $\left({\mathbb G, \oplus}\right)$ is an abelian group.

By Homomorphism Subgroup of All Mappings, it follows that $\left({\mathbb G, \oplus}\right)$ is a subgroup of $\left({G^G, \oplus}\right)$.


 * 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 the fact that the composition of mappings is associative.


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

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

Then:
 * $\left({u \oplus v}\right) * w = \left({u \oplus v}\right) \circ w$;
 * $u * \left({v \oplus w}\right) = u \circ \left({v \oplus w}\right)$.

So let $x \in G$. Then:

So $\left({u \oplus v}\right) * w = \left({u * w}\right) \oplus \left({v * w}\right)$.

Similarly:

So $u * \left({v \oplus w}\right) = \left({u * v}\right) \oplus \left({u * w}\right)$.

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


 * The ring axioms are satisfied, and $\left({\mathbb G, \oplus, *}\right)$ is a ring.


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


 * $e: G \to \left\{{e_G}\right\}: e \left({x}\right) = e_G$


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