Czechoslovak Mathematical Journal, first online, pp. 1-14


Maximal non valuation domains in an integral domain

Rahul Kumar, Atul Gaur

Received March 5, 2019.   Published online April 1, 2020.

Abstract:  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.
Keywords:  maximal non valuation domain; valuation subring; integrally closed subring
Classification MSC:  13B02, 13G05, 13F30, 13B22, 13B30
DOI:  10.21136/CMJ.2020.0098-19

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Affiliations:   Rahul Kumar, Atul Gaur (corresponding author), Department of Mathematics, University of Delhi, New Academic Block, University Enclave, Delhi, 110007, India, e-mail: rahulkmr977@gmail.com, gaursatul@gmail.com


 
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