Kernel of Ring Homomorphism is Subring

Theorem
Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ 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 of Group Homomorphism is Subgroup:
 * $\struct {\map \ker \phi}, +_1} \le \struct {R_1, +_1}$

where $\le$ denotes subgroup.

Let $x, y \in \map \ker \phi$.

Thus $x \circ_1 y \in \map \ker \phi$.

Thus the conditions for Subring Test are fulfilled, and $\map \ker \phi$ is a subring of $R_1$.