Definition:Local Ring Homomorphism/Definition 3

Definition
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 the preimage $f^{-1}(\mathfrak n) = \mathfrak m$.

Also see

 * Equivalence of Definitions of Local Ring Homomorphism