Lemma 10.18.4. Let $\varphi : R \to S$ be a ring map. Assume $R$ and $S$ are local rings. The following are equivalent:

$\varphi $ is a local ring map,

$\varphi (\mathfrak m_ R) \subset \mathfrak m_ S$, and

$\varphi ^{-1}(\mathfrak m_ S) = \mathfrak m_ R$.

For any $x \in R$, if $\varphi (x)$ is invertible in $S$, then $x$ is invertible in $R$.

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