Math and science::Algebra::Aluffi
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.
- \( \groupAdd{G} : G \times G \to G \) is the group operation of \( G \).
- \( \rho, \phi \in \operatorname{Endo}(G) \) are group endomorphisms.
- \( + : \operatorname{Endo}(G) \times \operatorname{Endo}(G) \to \operatorname{Endo}(G) \) is the derived group operation of the ring.
- \( a, b \in G \) are two elements of \( G \) and \( x = a \groupAdd{G} b \).
Requirement for \( + \)
The term \( \rho{\small{+}}\phi(x) \) decomposes in two ways, which must be equal.
Firstly, by definition of the new operation:
[\[
\begin{align*}
\rho{\small{+} }\phi(x) &= \rho(x) \groupAdd{G} \phi(x) \\
&= \;\; ? \groupAdd{G} \;?\; \groupAdd{G} \;?\; \groupAdd{G} \;? \end{align*}
\]]
And as \( + \) produces a group homomorphism:
[\[
\begin{align*}
\rho{\small{+} }\phi(x) &= \rho{\small{+} }\phi(a) \groupAdd{G} \rho{\small{+} }\phi(b) \\
&= \;\; ? \groupAdd{G} \;?\; \groupAdd{G} \;?\; \groupAdd{G} \;? \end{align*}
\]]
The requirement for these to decompositions to be equal is what restricts the construction to abelian groups.