Preimage of Prime Ideal under Ring Homomorphism is Prime Ideal

Theorem
Let $A$ and $B$ be commutative rings with unity.

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

Let $\mathfrak p \subseteq B$ be a prime ideal.

Then its preimage $\map {f^{-1} } {\mathfrak p}$ is a prime ideal of $A$.

Also see

 * Definition:Induced Mapping on Spectra of Rings