Fundamental Isomorphism Theorem (Ring)

definitionring-theory

Definition

For a morphism $f:(R,+,\cdot )\to (S,+,\cdot )$,

$${}^R{/}_{\ker f}\simeq \mathrm{im}f\text{ as rings}.$$

An isomorphism is $F(r+\ker f)=f(r)$.

Proof

Already know from groups that $F$ is a well-defined morphism of additive groups, and an isomorphism of groups (bijective). Hence we know it preserves addition, now it remains to show it preserves multiplication.

$$\begin{array}{rl}F((r+\ker f)(s+\ker f))& \stackrel{?}{=}F(r+\ker f)\cdot F(s+\ker f)\\ f((rs)+\ker f)& \stackrel{?}{=}f(r)\cdot f(s)\\ f(rs)& =f(rs).\end{array}$$