Commutative Algebra

$\phi( \mathfrak{m}) \subseteq \mathfrak{n}$

$\phi( \mathfrak{m}) = \mathfrak{n}$

$\phi^{-1}( \mathfrak{n}) = \mathfrak{m}$

$\phi$ induces a map of residue fields $A/ \mathfrak{m} \to B/ \mathfrak{n}$ compatible with $\phi$.

If $x$ is a unit of $A$, then $\phi(x)$ is a unit of $B$.

If $\phi(x)$ is a unit of $B$ for some $x\in A$, then $x$ is a unit of $A$.

$\phi$ is continuous with respect to the Krull topologies on $A$ and $B$, i.e. the $\mathfrak{m}$-adic topology on $A$ and $\mathfrak{n}$-adic topology on $B$.