Equivalence of Definitions of Local Ring Homomorphism

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Theorem

Let $(A, \mathfrak m)$ and $(B, \mathfrak n)$ 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 $f^{-1}(\mathfrak n) \supseteq \mathfrak m$.


Definition 3

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


Proof

1 iff 2

Follows from Image is Subset iff Subset of Preimage.

$\Box$


2 implies 3

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

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

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

By Proper Ideal of Ring is Contained in Maximal Ideal, $f^{-1}(\mathfrak n)$ 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.

$\blacksquare$