Definition:Contraction of Ideal

Definition

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

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

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

The contraction of $\mathfrak b$ by $f$ is its preimage under $f$:

$\mathfrak b^c = f^{-1} \sqbrk {\mathfrak b}$

Also see

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

Sources

• 1969: M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra ... (previous): Chapter $1$: Rings and Ideals: $\S$ Extension and Contraction