Ring of Endomorphisms is Ring with Unity

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Theorem

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

Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$.

Let $\struct {\mathbb G, \oplus, *}$ denote the ring of endomorphisms on $\struct {G, \oplus}$.


Then $\struct {\mathbb G, \oplus, *}$ is a ring with unity.


Proof

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


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By Set of Homomorphisms to Abelian Group 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:

\(\ds \map {\paren {\paren {u \oplus v} * w} } x\) \(=\) \(\ds \map {\paren {\paren {u \oplus v} \circ w} } x\)
\(\ds \) \(=\) \(\ds \map {\paren {u \oplus v} } {\map w x}\)
\(\ds \) \(=\) \(\ds \map u {\map w x} \oplus \map v {\map w x}\)
\(\ds \) \(=\) \(\ds \map {\paren {u \circ w} } x \oplus \map {\paren {v \circ w} } x\)
\(\ds \) \(=\) \(\ds \map {\paren {u * w} } x \oplus \map {\paren {v * w} } x\)

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

Similarly:

\(\ds \map {\paren {u * \paren {v \oplus w} } } x\) \(=\) \(\ds \map {\paren {u \circ \paren {v \oplus w} } } x\)
\(\ds \) \(=\) \(\ds \map u {\map {\paren {v \oplus w} } x}\)
\(\ds \) \(=\) \(\ds \map u {\map v x \oplus \map w x}\)
\(\ds \) \(=\) \(\ds \map u {\map v x} \oplus \map u {\map w x}\) $u$ has the morphism property
\(\ds \) \(=\) \(\ds \map {\paren {u \circ v} } x \oplus \map {\paren {u \circ w} } x\)
\(\ds \) \(=\) \(\ds \map {\paren {u * v} } x \oplus \map {\paren {u * w} } x\)

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.

$\blacksquare$


Also see


Sources

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