> [!NOTE] Lemma (First Isomorphism Theorem for Rings) \[MA268\] > Let $R_{1},R_{2}$ be [[Rings|rings]] and $\phi :R_{1}\to R_{2}$ a [[Homomorphism|homomorphism]]. Define$\begin{align}\hat{\phi}:R_{1}/ \text{Ker}(\phi)\to \text{Im}(\phi) \\(r+\text{Ker}(\phi))\mapsto \phi(r) \end{align}$where $\text{Ker}(\phi)$ denotes the [[Kernel of a Homomorphism of Rings|kernel]] of $\phi$, which [[Kernel of Homomorphism of Rings is an Ideal of Domain|is an ideal]] of $R_{1}$; $\text{Im}(\phi)$ denotes the [[Image of a Homomorphism of Rings|image]] of $\phi$; and $R_{1}/ \text{Ker}(\phi)$ denotes the [[Quotient Ring|quotient ring]] of $R_{1}$ modulo $\text{Ker}(\phi)$. Then $\phi$ is well-defined [[Isomorphism of Rings|isomorphism of rings]]. ###### Proof Analogous to [[First Isomorphism Theorem for Groups]].