Type conditions of stable range for identification of qualitative generalized classes of rings
This article deals mostly with the following question: when the classical ring of quotients of a commutative ring is a ring of stable range 1? 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, a semihereditary local rin...
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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Інститут прикладної математики і механіки НАН України
2018
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/188381 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Type conditions of stable range for identification of qualitative generalized classes of rings / B.V. Zabavsky // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 144–152 . — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862639513333923840 |
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| author | Zabavsky, B.V. |
| author_facet | Zabavsky, B.V. |
| citation_txt | Type conditions of stable range for identification of qualitative generalized classes of rings / B.V. Zabavsky // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 144–152 . — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | This article deals mostly with the following question: when the classical ring of quotients of a commutative ring is a ring of stable range 1? 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, a semihereditary local ring, a regular local ring. We find relationships between the introduced classes of rings and known ones, in particular, it is established that a commutative indecomposable almost clean ring is a regular local ring. Any commutative ring of idempotent regular range 1 is an almost clean ring. It is shown that any commutative indecomposable almost clean Bezout ring is an Hermite ring, any commutative semihereditary ring is a ring of idempotent regular range 1. The classical ring of quotients of a commutative Bezout ring QCl(R) is a (von Neumann) regular local ring if and only if R is a commutative semihereditary local ring.
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| first_indexed | 2025-12-01T01:32:42Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-188381 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-01T01:32:42Z |
| publishDate | 2018 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Zabavsky, B.V. 2023-02-26T12:38:04Z 2023-02-26T12:38:04Z 2018 Type conditions of stable range for identification of qualitative generalized classes of rings / B.V. Zabavsky // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 144–152 . — Бібліогр.: 6 назв. — англ. 1726-3255 2010 MSC: 13F99, 06F20. https://nasplib.isofts.kiev.ua/handle/123456789/188381 This article deals mostly with the following question: when the classical ring of quotients of a commutative ring is a ring of stable range 1? 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, a semihereditary local ring, a regular local ring. We find relationships between the introduced classes of rings and known ones, in particular, it is established that a commutative indecomposable almost clean ring is a regular local ring. Any commutative ring of idempotent regular range 1 is an almost clean ring. It is shown that any commutative indecomposable almost clean Bezout ring is an Hermite ring, any commutative semihereditary ring is a ring of idempotent regular range 1. The classical ring of quotients of a commutative Bezout ring QCl(R) is a (von Neumann) regular local ring if and only if R is a commutative semihereditary local ring. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Type conditions of stable range for identification of qualitative generalized classes of rings Article published earlier |
| spellingShingle | Type conditions of stable range for identification of qualitative generalized classes of rings Zabavsky, B.V. |
| title | Type conditions of stable range for identification of qualitative generalized classes of rings |
| title_full | Type conditions of stable range for identification of qualitative generalized classes of rings |
| title_fullStr | Type conditions of stable range for identification of qualitative generalized classes of rings |
| title_full_unstemmed | Type conditions of stable range for identification of qualitative generalized classes of rings |
| title_short | Type conditions of stable range for identification of qualitative generalized classes of rings |
| title_sort | type conditions of stable range for identification of qualitative generalized classes of rings |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/188381 |
| work_keys_str_mv | AT zabavskybv typeconditionsofstablerangeforidentificationofqualitativegeneralizedclassesofrings |