I have to show the following proposition:
Let $O$ be a commutative unitary ring that contains $Z$ the ring of integers. $I\le O$ an ideal. Suppose that $\frac{O}{I}\cong\frac{Z}{pZ}$ with p prime integer. Then, $O=Z+I$.
My idea was: let $\varphi$ be the isomorphism, take $Z+I$ in $\frac{O}{I}$ by the isomorphism $\varphi(Z+I)=\frac{Z}{pZ}$, then $Z+I=\varphi^{-1}(\frac{Z}{pZ})=\frac{O}{I}$. By the correspondence theorem, the thesis follows.
If my idea is correct, how would i show that $\varphi(Z+I)=\frac{Z}{pZ}$? suggestions?
[The proposition is taken from an exercise, so it may miss some information. Tell me if needed other hypothesis]. thank you.