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 $A$.

Proof

Let $x,y \in A$ such that:

$xy \in \mathfrak b^c$.

That is:

$\map f {xy} = \map f x \map f y \in \mathfrak b$

Suppose that $x \ne \mathfrak b^c$.

That is:

$\map f x \not \in \mathfrak b$

Since $\mathfrak b$ is primary:

$\exists n \in \N_{>0} : \map f {y^n} = \paren {\map f y}^n \in \mathfrak b$

That is:

$y^n \in \mathfrak b^c$

$\blacksquare$