\begin{definition}[Definition:Extension of Ideal]
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 $f$:
:$\mathfrak a^e = \left\langle f(\mathfrak a) \right\rangle$
\end{definition}