Jacobson Radical in Localization.

Question

Let $A$ a ring comutative with 1 and $I\subset A$ an ideal. Set $S=1+I$ a multiplicative set. Show that $S^{-1}I$ is contained Jacobson Radical of $S^{-1}A$.
Hint?

Answer 1

Hint

You aim to show that $S^{-1}I$ is contained in every maximal ideal of $S^{-1}A$. Recall from the ideal correspondence of localizations that the maximal ideals of $S^{-1}A$ are the ideals $S^{-1} \mathfrak{m}$ where $\mathfrak{m}$ is an ideal of $A$ maximal with respect to being disjoint from $S$. It's probably a good start to try to characterize these ideals!

Check spoiler below if you need help. Proving the characterization is straightforward.

An ideal $\mathfrak{m}$ of $A$ is maximal with respect to being disjoint from $S = 1 + I$ iff $\mathfrak{m}$ is a maximal ideal of $A$ and $\mathfrak{m} \supseteq I$.