endomorphism ring of an abelian group

follows from abelian group homomorphisms as abelian group

follows from composition of group homomorphisms as group homomorphism, identity morphism as neutral element.

Let \(\varphi_1,\varphi_2,\varphi_3\) be endomorphisms, then for arbitrary \(g \in G\) it holds, that

\begin{align*} \varphi_1 \cdot (\varphi_2 + \varphi_3)(g) =& \varphi_1 \circ (\varphi_2(g) + \varphi_3(g)) \\ =& \varphi_1(\varphi_2(g)) + \varphi_1(\varphi_3(g)) \\ =& (\varphi_1 \circ \varphi_2)(g) + (\varphi_1 \circ \varphi_2)(g) \\ =& ((\varphi_1 \cdot \varphi_2) + (\varphi_1 \cdot \varphi_3))(g) \end{align*}

and

\begin{align*} (\varphi_2 + \varphi_3) \cdot \varphi_1(g) =& (\varphi_2 + \varphi_3) \circ \varphi_1(g) \\ =& \varphi_2(\varphi_1(g)) + \varphi_3(\varphi_1(g)) \\ =& (\varphi_2 \circ \varphi_1)(g) + (\varphi_3 \circ \varphi_1)(g) \\ =& (\varphi_2 \cdot \varphi_1 + \varphi_3 \cdot \varphi_1)(g) \\ \end{align*}