Equivalence of Definitions of Local Ring Homomorphism

Theorem

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 following definitions of the concept of Local Ring Homomorphism are equivalent:

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 $\map {f^{-1} } {\mathfrak n} = \mathfrak m$.

Proof

1 iff 2

Follows from Image is Subset iff Subset of Preimage.

$\Box$

2 implies 3

Let $f^{-1} \sqbrk {\mathfrak n} \supseteq \mathfrak m$.

We have to show that $f^{-1} \sqbrk {\mathfrak n} \subseteq \mathfrak m$.

By Preimage of Proper Ideal of Ring is Proper Ideal, $f^{-1} \sqbrk {\mathfrak n}$ is a proper ideal.

By Proper Ideal of Ring is Contained in Maximal Ideal, $f^{-1} \sqbrk {\mathfrak n}$ is contained in some maximal ideal of $A$.

Because $A$ is a commutative local ring, $\mathfrak m$ is its only maximal ideal.

$\Box$

3 implies 2

Follows by definition of set equality.

$\blacksquare$