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 is Subring, $$\mathrm{ker} \left({\phi}\right)$$ is a subring of $$R_1$$.


 * Let $$s \in \mathrm{ker} \left({\phi}\right)$$, 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.