Ring. As a group of group endomorphisms.

A fundamental way to motivate rings is via endomorphisms of a group.

A requirement of this construction is that the original group be abelian. The reason is captured below.

Setup

• $G$ is a group.
• ${+}_G:G\times G\to G$ is the group operation of $G$.
• $\rho ,\varphi \in \mathrm{Endo}(G)$ are group endomorphisms.
• $+:\mathrm{Endo}(G)\times \mathrm{Endo}(G)\to \mathrm{Endo}(G)$ is the derived group operation of the ring.
• $a,b\in G$ are two elements of $G$ and $x=a{+}_{G}b$.

Requirement for $+$

The term $\rho +\varphi (x)$ decomposes in two ways, which must be equal.

Firstly, by definition of the new operation:

And as $+$ produces a group homomorphism:

The requirement for these to decompositions to be equal is what restricts the construction to abelian groups.