Extension and contraction of a ring

Last updated January 27, 2022

Definition

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

Definition. Given an ideal $\mathfrak{a}\subset A$, its extension under $f$ is the ideal in $B$ generated by its image: $$\mathfrak{a}^e$$= (f(\mathfrak{a}))\subset B.$

Definition. Given an ideal $\mathfrak{b}\subset B, its contraction under $f$ is the ideal which is precisely the preimage of $\mathfrak{b}$: $$\mathfrak{b}^c=f^{-1}(\mathfrak{b})\subset A.$$

Definition. An extended ideal

Properties

1. The contraction of a prime ideal is always prime, but the extension of a prime ideal need not be prime.