\begin{definition}[Definition:Local Ring Homomorphism/Definition 2]
Let $(A, \mathfrak m)$ and $(B, \mathfrak n)$ be commutative local rings.
Let $f : A \to B$ be a unital ring homomorphism.
The homomorphism $f$ is '''local''' {{iff}} the preimage $f^{-1}(\mathfrak n) \supseteq \mathfrak m$.
\end{definition}