TITLE: Prove: The pre-image of an ideal is an ideal.
QUESTION [10 upvotes]: Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$.

REPLY [31 votes]: it is the kernel of $R \rightarrow S \rightarrow S/N$