# A question about integral domains and inclusion relations in abstract algebra

Let $R$ be a Prüfer domain with quotient field $K$ and $\sum$ be the set of all semilocal Prüfer domains $R'$ with $n$ maximal ideals and quotient field $K$ such that $R\subseteq R'$. Let $R=R'_{1}\cap\ldots\cap R'_{t}$ where $R'_{1},\ldots,R'_{t}$ are minimal elements of $\sum$ (ordered by inclusion) and pairwise incomparable. Is $\{R'_{1},\ldots,R'_{t}\}$ the set of all minimal elements of $\sum$ ?

Thanks for your answers.

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I vote to keep this question open. –  user18119 Jan 15 '13 at 22:54
Let me ask this question in general case: Let $R$ be a integral domain with quotient field $K$. Let $\sum$ be the set of all overrings of R (a subring of $K$ containing $R$). Assume that $R=R_{1}\cap\ldots\cap R_{t}$ ($t$ is finite) where $R_{1},\ldots,R_{t}$ are minimal elements of $\sum$ (ordered by inclusion) and $R_{1},\ldots,R_{t}$ are pairwise incomparable by inclusion. Is $\{R_{1},\ldots,R_{t}\}$ the set of all minimal elements of $\sum$ ? –  Azadeh Jan 16 '13 at 10:05