Contraction of an ideal

Definition

Let $f:R\to S$ be a homomorphism of commutative unital rings. Given an ideal $I$ in $S$, the contraction of $I$ to $R$ is the full inverse image $f^{-1}(I)$. When the map $f:R\to S$ is understood, we denote the contraction simple as $I^c$.

The contraction of an ideal is always an ideal.