Let $R$ be a commutative ring with 1, and let $0\not=a$ be an element of $R$ such that there exists an ideal $I$ of $R$ with the property that $a\not\in I$ and if $J$ is an ideal with $a\not\in J$ then $J\subseteq I$. Set $$\mathfrak{m}:=\{r\in R\mid ra\in I\}$$ and assume that $\mathfrak{m}$ is a maximal ideal and an essential ideal of $R$ . How can we show that $$\bigcap_{\mathfrak{n}\in \mathrm{Max}(R)\smallsetminus\{\mathfrak{m}\}}\mathfrak{n}\not\subseteq \mathfrak{m},$$ where $\mathrm{Max}(R)$ is the set of all maximal ideals of $R$?

Note: $\bigcap\varnothing=R$ and an essential ideal is an ideal that has nonzero intersection with every nonzero ideal.


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