Definition:Local Ring Homomorphism

Definition
Let $A$ and $B$ be local commutative rings with unity with maximal ideals $\mathfrak m$ and $\mathfrak n$.

Let $f : A \to B$ be a ring homomorphism.

Definition 1
The homomorphism $f$ is local $f$ is unital and the image $f(\mathfrak m) \subseteq \mathfrak n$.

Definition 2
The homomorphism $f$ is local $f$ is unital and the preimage $f^{-1}(\mathfrak n) = \mathfrak m$.

Definition 3
The homomorphism $f$ is local $f$ is unital and the preimage $f^{-1}(\mathfrak n) \supseteq \mathfrak m$.

Also see

 * Definition:Category of Local Rings