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.