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Manis valuations and Prüfer extensions

Manfred Knebusch (1998)

Banach Center Publications

Manis valuations and Prüfer extensions. I.

Knebusch, Manfred, Zhang, Digen (1996)

Documenta Mathematica

Maximal non valuation domains in an integral domain

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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...

Maximal non $λ$ -subrings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

Let $R$ be a commutative ring with unity. The notion of maximal non $λ$ -subrings is introduced and studied. A ring $R$ is called a maximal non $λ$ -subring of a ring $T$ if $R \subset T$ is not a $λ$ -extension, and for any ring $S$ such that $R \subset S \subseteq T$ , $S \subseteq T$ is a $λ$ -extension. We show that a maximal non $λ$ -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 $λ$ -domain is given. A necessary condition is given...

Maximal non-Jaffard subrings of a field.

Mabrouk Ben Nasr, Noôman Jarboui (2000)

Publicacions Matemàtiques

A domain R is called a maximal non-Jaffard subring of a field L if R ⊂ L, R is not a Jaffard domain and each domain T such that R ⊂ T ⊆ L is Jaffard. We show that maximal non-Jaffard subrings R of a field L are the integrally closed pseudo-valuation domains satisfying dimv R = dim R + 1. Further characterizations are given. Maximal non-universally catenarian subrings of their quotient fields are also studied. It is proved that this class of domains coincides with the previous class when R is integrally...

Maximal non-pseudovaluation subrings of an integral domain

Rahul Kumar (2024)

Czechoslovak Mathematical Journal

The notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let $R \subset S$ be an extension of domains. Then $R$ is called a maximal non-pseudovaluation subring of $S$ if $R$ is not a pseudovaluation subring of $S$ , and for any ring $T$ such that $R \subset T \subset S$ , $T$ is a pseudovaluation subring of $S$ . We show that if $S$ is not local, then there no such $T$ exists between $R$ and $S$ . We also characterize maximal non-pseudovaluation subrings of a local integral domain.