> [!NOTE] Lemma (Kernel of Homomorphism of Rings is an Ideal of Domain) > Let $R_{1}$ and $R_{2}$ be [[Rings|rings with unity]] and $\phi: R_{1} \to R_{2}$ be a [[Homomorphism of Rings|homomorphism]]. Then its [[Kernel of a Homomorphism of Rings|kernel]] is an [[Ideal of Ring|ideal]] of $R_{1}$. **Proof** It follows from [[Kernel of Homomorphisms of Groups is Normal Subgroup of Domain]] that $\text{Ker}(\phi)$ is a subgroup of $(R_{1},+)$. The fact that $\text{Ker}(\phi)$ is an ideal follows from [[Ring Product With Zero is Zero]].