# Definition:Extension of Ideal

## Definition

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

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

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

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

$\mathfrak a^e = \left\langle f(\mathfrak a) \right\rangle$