Extension and contraction of ideals

For a ring homomorphism $f:R\to S$, extension sends an ideal $I\subseteq R$ to the ideal $Sf(I)$ generated by its image. Contraction sends an ideal $J\subseteq S$ to $f^{-1}(J)$. These are distinct forward and inverse-image operations.

For a ring homomorphism $f:R\to A$ and an ideal $I\subseteq R$, the extension $I^e=IA$ is the ideal of $A$ generated by $f(I)$.

For a ring homomorphism $f:R\to A$ and an ideal $J\subseteq A$, its contraction is $J^c=f^{-1}(J)$.