Preimage of Image of Ideal under Ring Homomorphism

Theorem
Let $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$ be a ring epimorphism.

Let $K = \ker \left({\phi}\right)$, where $\ker \left({\phi}\right)$ is the kernel of $\phi$.

Let $J$ be an ideal of $R_1$.

Then:


 * $\phi^{-1} \left({\phi \left({J}\right)}\right) = J + K$

Proof
As an ideal is a subring, the result Ring Epimorphism Composite with Inverse of Subring applies directly.