Preimage of Ideal under 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 epimorphism.

Let $$S_2$$be an ideal of $$R_2$$.

Then $$S_1 = \phi^{-1} \left({S_2}\right)$$ is an ideal of $$R_1$$ such that $$\ker \left({\phi}\right) \subseteq S_1$$.

Proof
Uses a similar argument to Ring Epimorphism Composite with Inverse of Subring.