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Keywords:
maximal non valuation domain; valuation subring; integrally closed subring
maximal non valuation domain; valuation subring; integrally closed subring
Summary:
Let $R$ be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring $R$ of an integral domain $S$ is called a maximal non valuation domain in $S$ if $R$ is not a valuation subring of $S$, and for any ring $T$ such that $R \subset T\subset S$, $T$ is a valuation subring of $S$. For a local domain $S$, the equivalence of an integrally closed maximal non VD in $S$ and a maximal non local subring of $S$ is established. The relation between $\dim (R,S)$ and the number of rings between $R$ and $S$ is given when $R$ is a maximal non VD in $S$ and $\dim (R,S)$ is finite. For a maximal non VD $R$ in $S$ such that $R\subset R^{\prime _S} \subset S$ and $\dim (R,S)$ is finite, the equality of $\dim (R,S)$ and $\dim (R^{\prime _S},S)$ is established.
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