Definition:Local Ring Homomorphism
Jump to navigation
Jump to search
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
- Results about local ring homomorphisms can be found here.