Definition:Extension of Ideal

This page is about extensions of ideals. For other uses, see Extension.

Definition

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

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

Let $\mathfrak a$ be an ideal of $A$.

The extension of $\mathfrak a$ by $f$ is the ideal generated by its image under $f$:

$\mathfrak a^e = \left\langle f \sqbrk {\mathfrak a} \right\rangle$

Also see

• Definition:Contraction of Ideal

Sources

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