Equivalence of Definitions of Local Ring Homomorphism

Theorem
Let $(A, \mathfrak m)$ and $(B, \mathfrak n)$ be commutative local rings.

Let $f : A \to B$ be a unital ring homomorphism.

1 iff 2
Follows from Image is Subset iff Subset of Preimage.

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.

3 implies 2
Follows by definition of set equality.