Kernel of Ring Homomorphism is Subring

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 a subring of $R_1$.

Proof
From Ring Homomorphism of Addition is Group Homomorphism and Kernel is Subgroup:
 * $\left({\ker \left({\phi}\right), +_1}\right) \le \left({R_1, +_1}\right)$.

where $\le$ denotes subgroup.

Let $x, y \in \ker \left({\phi}\right)$.

Thus $x \circ_1 y \in \ker \left({\phi}\right)$.

Thus the conditions for Subring Test are fulfilled, and $\ker \left({\phi}\right)$ is a subring of $R_1$.