$\phi:R\to S$ is a ring homomorphism and $I\subset S$ is an ideal, then $\phi$ induces an injection $R/(\phi^{-1}(I))\subset S/I$.

Question

This is a statement made in Eisenbud Commutative Algebra Chapter 2, localization Section 1.

If $\phi:R\to S$ is any map of sets, then operation taking subs of $S$ to subsets of $R$ by $I\to \phi^{-1}(I)$ preserves inclusion and intersections. If $\phi$ is a map of rings, and $I\subset S$ is an ideal. Then he concludes $\phi$ induces an injection $R/\phi^{-1}(I)\subset S/I$.

I am not clear about this injection here. I guess he means injection in the sense of ideals rather than elements.

Answer 1

Consider the canonical projection $\pi: S \rightarrow S/I$. The kernel of $\pi \circ \phi$ is $\phi^{-1}(I)$, so by the homomorphism theorem, there is an induced injection $R/\phi^{-1}(I) \rightarrow S/I$.