Preimage of Image of Ideal under Ring Homomorphism

Theorem
Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism.

Let $K = \map \ker \phi$, be the kernel of $\phi$.

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

Then:


 * $\phi^{-1} \sqbrk {\phi \sqbrk J} = J + K$

Proof
As an ideal is a subring, the result Preimage of Image of Subring under Ring Homomorphism applies directly.