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.