Definition:Contraction of Ideal

Definition
Let $A$ and $B$ be commutative ring with unity.

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

Let $\mathfrak b \subseteq B$ be an ideal.

The contraction of $\mathfrak b$ by $f$ is its preimage under $f$:
 * $\mathfrak b^c = f^{-1}(\mathfrak b)$

Also see

 * Preimage of Ideal under Ring Homomorphism is Ideal
 * Definition:Extension of Ideal