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november 2020 A note on $\lambda$-domains and $\Delta$-domains
Rahul Kumar, Atul Gaur
Bull. Belg. Math. Soc. Simon Stevin 27(4): 499-508 (november 2020). DOI: 10.36045/j.bbms.190718

Abstract

Let $R$ be an integral domain. Then $R$ is said to be a $\lambda$-domain if the set of all overrings of $R$ is linearly ordered by inclusion. If $R_1 + R_2$ is an overring of $R$ for each pair of overrings $R_1, R_2$ of $R$, then $R$ is said to be a $\Delta$-domain. We show that if $R\subset T$ is an extension of integral domains such that each proper subring of $T$ containing $R$ is a $\lambda$-domain (resp., $\Delta$-domain), then $T$ is a $\lambda$-domain (resp., $\Delta$-domain under some conditions). Moreover, the pair $(R, T)$ is a residually algebraic pair. Two new ring theoretic properties, namely $\lambda$-property of domains and $\Delta$-property of domains are introduced and studied.

Citation

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Rahul Kumar. Atul Gaur. "A note on $\lambda$-domains and $\Delta$-domains." Bull. Belg. Math. Soc. Simon Stevin 27 (4) 499 - 508, november 2020. https://doi.org/10.36045/j.bbms.190718

Information

Published: november 2020
First available in Project Euclid: 20 November 2020

MathSciNet: MR4177389
Digital Object Identifier: 10.36045/j.bbms.190718

Subjects:
Primary: 13B02 , 13G05
Secondary: 13A18 , 13G99

Keywords: $\Delta$-domain , $\lambda$-domain , Maximal subring , Overring , Valuation domain

Rights: Copyright © 2020 The Belgian Mathematical Society

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Vol.27 • No. 4 • november 2020
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