Contraction of Primary Ideal is Primary Ideal

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

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

Let $\mathfrak b$ be a primary ideal of $B$.

Let $\mathfrak b^c$ be the contraction of $\mathfrak b$ by $f$:

Then $\mathfrak b^c$ is a primary ideal of $A$.