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 if and only if the image $f(\mathfrak m) \subseteq \mathfrak n$.

Also see

• Equivalence of Definitions of Local Ring Homomorphism

Sources

• 1972: N. Bourbaki: Commutative Algebra ... (previous) Chapter $\text {II}$: Localization: $\S3$ Local rings. Passage from the local to the global $1$: Local rings: Definition $2$