Stable range conditions for abelian and duo rings
The article deals with the following question: when does the classical ring of quotients of a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are there idempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regular range 1, a ring of se...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2022-03-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/244 |
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| Summary: | The article deals with the following question: when does the classical ring of quotients
of a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are there
idempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regular
range 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationships
between 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. We
proved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3). |
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| ISSN: | 1027-4634 2411-0620 |