Kernel of Ring Homomorphism is Ideal

Theorem
Let $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$ be a ring homomorphism.

Then the kernel of $\phi$ is an ideal of $R_1$.

Proof
By Kernel of Ring Homomorphism is Subring, $\ker \left({\phi}\right)$ is a subring of $R_1$.

Let $s \in \ker \left({\phi}\right)$, so $\phi \left({s}\right) = 0_{R_2}$.

Suppose $x \in R_1$. Then:

and similarly for $\phi \left({s \circ_1 x}\right)$.

The result follows.