Maximal non valuation domains in an integral domain

Kumar, Rahul, and Gaur, Atul. "Maximal non valuation domains in an integral domain." Czechoslovak Mathematical Journal 70.4 (2020): 1019-1032. <http://eudml.org/doc/297239>.

@article{Kumar2020,
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.},
author = {Kumar, Rahul, Gaur, Atul},
journal = {Czechoslovak Mathematical Journal},
keywords = {maximal non valuation domain; valuation subring; integrally closed subring},
language = {eng},
number = {4},
pages = {1019-1032},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Maximal non valuation domains in an integral domain},
url = {http://eudml.org/doc/297239},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Kumar, Rahul
AU - Gaur, Atul
TI - Maximal non valuation domains in an integral domain
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 4
SP - 1019
EP - 1032
AB - 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.
LA - eng
KW - maximal non valuation domain; valuation subring; integrally closed subring
UR - http://eudml.org/doc/297239
ER -