This question is taken from "Graduate Course in Algebra for Martin Isaacs" which asking whether a statement is true or not. The question says if $S \subset R$ is a unitary subring (so it contains the same unity $1$ as in $R$), then determine whether the following statements is true or not:
(a) $J(S) \subset J(R)$
(b) $J(R) \cap S\subset J(S)$.
Here, $J(S)$ is the intersection of all maximal ideals in $R$. So I think (a) is wrong because the passing the intersection reverse the inclusion, right? I don't know if I am right and if so, is there a counterexample support this?
In (b), I believe it is true, as the inclusion reversed.
a) Consider any local domain $S$ with nonzero maximal ideal, and let $R$ its field of fractions.
b) Consider $S=F[x]$ and $R=F[x]_{(x)}$ ($F[x]$ localized at the prime ideal $(x)$).