Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$. Now let $J=\cap_{I\subseteq m\in Max(R)}m$. Set $A:=\{p\in Spec(R): I\subseteq p\}$ and $B:=\{p\in Spec(R): J\subseteq p\}$, where $Spec(R)$ and $Max(R)$ are the set of all prime and maximal ideals of $R$ respectively. When considering $Spec(R)$ with the Zariski topology, and $B$ and $A$ as two subspaces of $Spec(R)$.I want if $B$ is dense in $A$? Or in general, what kinds of topological properties hold between $A$ and $B$?

  • With this degree of generality, essentially nothing: take for $R$ a local ring with maximal ideal $\mathfrak{m}$, and $I=0$. Then $A=\operatorname{Spec}(R) $ and $B=\{\mathfrak{m}\} $. – abx Sep 13 at 17:39

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