Preimage of Image of Subring under Ring Homomorphism

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

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

Let $J$ be a subring of $R_1$.

Then:


 * $\phi^{-1} \left[{\phi \left[{J}\right]}\right] = J + K$

Proof
Let $x \in \phi^{-1} \left[{\phi \left[{J}\right]}\right]$.

Then:

So we have shown that:
 * $\phi^{-1} \left[{\phi \left[{J}\right]}\right] \subseteq J + K$

Now suppose that $x \in J + K$.

Then:

So we have shown that:
 * $J + K \subseteq \phi^{-1} \left[{\phi \left[{J}\right]}\right]$

Hence the result by definition of set equality.