Maximal non $\lambda$-subrings

Kumar, Rahul, and Gaur, Atul. "Maximal non $\lambda $-subrings." Czechoslovak Mathematical Journal 70.2 (2020): 323-337. <http://eudml.org/doc/297132>.

@article{Kumar2020,
abstract = {Let $R$ be a commutative ring with unity. The notion of maximal non $\lambda $-subrings is introduced and studied. A ring $R$ is called a maximal non $\lambda $-subring of a ring $T$ if $R\subset T$ is not a $\lambda $-extension, and for any ring $S$ such that $R\subset S\subseteq T$, $S\subseteq T$ is a $\lambda $-extension. We show that a maximal non $\lambda $-subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D + M$ construction is a maximal non $\lambda $-domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non $\lambda $-subring. If $R$ is a maximal non $\lambda $-subring of a field $K$, where $R$ is integrally closed in $K$, then $K$ is the quotient field of $R$ and $R$ is a Prüfer domain. The equivalence of a maximal non $\lambda $-domain and a maximal non valuation subring of a field is established under some conditions. We also discuss the number of overrings, chains of overrings, and the Krull dimension of maximal non $\lambda $-subrings of a field.},
author = {Kumar, Rahul, Gaur, Atul},
journal = {Czechoslovak Mathematical Journal},
keywords = {maximal non $\lambda $-subring; $\lambda $-extension; integrally closed extension; valuation domain},
language = {eng},
number = {2},
pages = {323-337},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Maximal non $\lambda $-subrings},
url = {http://eudml.org/doc/297132},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Kumar, Rahul
AU - Gaur, Atul
TI - Maximal non $\lambda $-subrings
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 2
SP - 323
EP - 337
AB - Let $R$ be a commutative ring with unity. The notion of maximal non $\lambda $-subrings is introduced and studied. A ring $R$ is called a maximal non $\lambda $-subring of a ring $T$ if $R\subset T$ is not a $\lambda $-extension, and for any ring $S$ such that $R\subset S\subseteq T$, $S\subseteq T$ is a $\lambda $-extension. We show that a maximal non $\lambda $-subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D + M$ construction is a maximal non $\lambda $-domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non $\lambda $-subring. If $R$ is a maximal non $\lambda $-subring of a field $K$, where $R$ is integrally closed in $K$, then $K$ is the quotient field of $R$ and $R$ is a Prüfer domain. The equivalence of a maximal non $\lambda $-domain and a maximal non valuation subring of a field is established under some conditions. We also discuss the number of overrings, chains of overrings, and the Krull dimension of maximal non $\lambda $-subrings of a field.
LA - eng
KW - maximal non $\lambda $-subring; $\lambda $-extension; integrally closed extension; valuation domain
UR - http://eudml.org/doc/297132
ER -