Extension of an ideal

Definition
Let $$f:R \to S$$ is a homomorphism of commutative unital rings. Let $$I$$ be an ideal inside $$R$$. The extension of $$I$$ to $$S$$ is defined as the ideal of $$S$$ generated by the set-theoretic image $$f(I)$$. When the map is understood, we denote the extension by $$I^e$$.