Definition:Local Ring Homomorphism

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.

Definition 1

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

Definition 2

The homomorphism $f$ is local if and only if the preimage $\map {f^{-1} } {\mathfrak n} \supseteq \mathfrak m$.

Definition 3

The homomorphism $f$ is local if and only if the preimage $\map {f^{-1} } {\mathfrak n} = \mathfrak m$.

Also see

• Equivalence of Definitions of Local Ring Homomorphism
• Definition:Category of Local Rings

• Results about local ring homomorphisms can be found here.