Why quotient ring of intersection of annihilators for a Jacobson ring is Jacobson?

Question

Let $R$ be a commutative ring with identity and $M_i$ is a finitely generated $R$-module, for $i=1,\dots,n$. If for every $i$, $R/\operatorname{Ann}(M_i)$ is a Jacobson ring, why $R/\bigcap_{i=1}^n \operatorname{Ann}(M_i)$ is a Jacobson ring?

Answer 1

Let $I_1,\dots,I_n$ be ideals of $R$ such that $R/I_i$ is a Jacobson ring for every $i$. Let $P$ be a prime ideal of $R$ containing $I=\cap_{i=1}^n I_i$. Then there is $j$ such that $P\supseteq I_j$. Since $R/I_j$ is Jacobson we have $J(\dfrac{R/I_j}{P/I_j})=0$, so $J(R/P)=0$, and thus $J(\dfrac{R/I}{P/I})=0$.