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, $$\left({\mathrm{ker} \left({\phi}\right), +_1}\right) \le \left({R_1, +_1}\right)$$.

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

$$ $$ $$

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

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