Ring of Endomorphisms

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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 Structure Induced by Group Operation is Group, $\left({\mathbb G, \oplus}\right)$ is an abelian group.

By Set of Homomorphisms is 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 [Definition:Associative Operation|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:

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


So let $x \in G$.

Then:

\(\displaystyle \left({\left({u \oplus v}\right) * w}\right) \left({x}\right)\) \(=\) \(\displaystyle \left({\left({u \oplus v}\right) \circ w}\right) \left({x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({u \oplus v}\right) \left({w \left({x}\right)}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle u \left({w \left({x}\right)}\right) \oplus v \left({w \left({x}\right)}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({u \circ w}\right) \left({x}\right) \oplus \left({v \circ w}\right) \left({x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({u * w}\right) \left({x}\right) \oplus \left({v * w}\right) \left({x}\right)\)

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

Similarly:

\(\displaystyle \left({u * \left({v \oplus w}\right)}\right) \left({x}\right)\) \(=\) \(\displaystyle \left({u \circ \left({v \oplus w}\right)}\right) \left({x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle u \left({\left({v \oplus w}\right) \left({x}\right)}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle u \left({v \left({x}\right) \oplus w \left({x}\right)}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle u \left({v \left({x}\right)}\right) \oplus u \left({w \left({x}\right)}\right)\) $u$ has the morphism property
\(\displaystyle \) \(=\) \(\displaystyle \left({u \circ v}\right) \left({x}\right) \oplus \left({u \circ w}\right) \left({x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({u * v}\right) \left({x}\right) \oplus \left({u * w}\right) \left({x}\right)\)

So:

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

Thus $*$ 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.

$\blacksquare$


Notes

Note that $\left({\mathbb G, \oplus, *}\right)$ as defined here is not necessarily a commutative ring.


Sources