# When is a maximal ideal isolated?

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.