Spring 2022 Module-theoretic characterizations of the ring of finite fractions of a commutative ring
Fang Gui Wang, De Chuan Zhou, Dan Chen
J. Commut. Algebra 14(1): 141-154 (Spring 2022). DOI: 10.1216/jca.2022.14.141

Abstract

Let R be a commutative ring with identity and let 𝒬 be the set of finitely generated semiregular ideals of R. A 𝒬-torsion-free R-module M is called a Lucas module if ExtR1(RJ,M)=0 for any J𝒬. Moreover, R is called a DQ ring if every ideal of R is a Lucas module. We prove that if the small finitistic dimension of R is zero, then R is a DQ ring. In terms of a trivial extension, we construct a total ring of quotients of the type R=DH which is not a DQ ring. Thus in this case, the small finitistic dimension of R is not zero. This provides a negative answer to an open problem posed by Cahen et al.

Citation

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Fang Gui Wang. De Chuan Zhou. Dan Chen. "Module-theoretic characterizations of the ring of finite fractions of a commutative ring." J. Commut. Algebra 14 (1) 141 - 154, Spring 2022. https://doi.org/10.1216/jca.2022.14.141

Information

Received: 13 May 2019; Revised: 22 December 2019; Accepted: 26 December 2019; Published: Spring 2022
First available in Project Euclid: 31 May 2022

MathSciNet: MR4430705
zbMATH: 1496.13004
Digital Object Identifier: 10.1216/jca.2022.14.141

Subjects:
Primary: 13A15 , 13C99

Keywords: Lucas module , semiregular ideal , small finitistic dimension , the ring of finite fractions

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

Vol.14 • No. 1 • Spring 2022
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