阿贝尔环和二重环的稳定值域条件

Q3 Mathematics Matematychni Studii Pub Date : 2022-03-31 DOI:10.30970/ms.57.1.92-97
A. Dmytruk, A. Gatalevych, M. Kuchma
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引用次数: 1

摘要

本文讨论了如下问题:对偶环的经典商环何时存在,商环$Q_{Cl}(R)$中的幂等元在$R$中是幂等元?在本文中,我们引入了(von Neumann)正则范围1的环、半遗传范围1的圈、正则范围1环的概念。我们发现了引入的环类与阿贝尔环和对偶环的已知环类之间的关系。证明了半遗传局部对偶环是半遗传范围为1的环。还证明了正则局部Bezout对偶环是一个稳定范围为2的环。特别地,证明了以下定理1:对于阿贝尔环$R$,以下条件是等价的:$1.$\$R$是稳定范围为1的环$2.$\$R$是冯-诺依曼正则范围1的一个环。本文还引入了Gelfand元素的概念以及对偶环的Gelfand范围为1的环。我们证明了Gelfand范围1的Hermite对偶环是初等除数环(定理3)。
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Stable range conditions for abelian and duo rings
The article deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$\ $R$ is a ring of stable range 1; $2.$\ $R$ is a ring of von Neumann regular range 1. The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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