First Fundamental Theorem on Ring Homomorphisms

Theorem
Let $R$ and $S$ be rings and $\phi:R\mapsto S$ be a ring homomorphism.

Then
 * $S\cong R/\operatorname{Ker}(\phi)$,

where $\operatorname{Ker}(\phi) = \{r\in R : \phi(r)=0_S\}$ is the kernel of $\phi$.