Definition:Local Ring Homomorphism/Definition 1

Definition
Let $\struct {A, \mathfrak m}$ and $\struct {B, \mathfrak n}$ be commutative local rings.

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

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

Also see

 * Equivalence of Definitions of Local Ring Homomorphism