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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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/503 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543352929910784 |
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| author | Zabavsky, Bohdan Volodymyrovych |
| author_facet | Zabavsky, Bohdan Volodymyrovych |
| author_sort | Zabavsky, Bohdan Volodymyrovych |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-10-20T08:02:25Z |
| 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 \(Q_{Cl}(R)\) is a (von Neumann) regular local ring if and only if \(R\) is a commutative semihereditary local ring. |
| first_indexed | 2026-02-08T08:01:52Z |
| format | Article |
| id | admjournalluguniveduua-article-503 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T08:01:52Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-5032018-10-20T08:02:25Z Type conditions of stable range for identification of qualitative generalized classes of rings Zabavsky, Bohdan Volodymyrovych Bezout ring, Hermite ring, elementary divisor ring, semihereditary ring, regular ring, neat ring, clean ring, stable range 1 13F99, 06F20 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 \(Q_{Cl}(R)\) is a (von Neumann) regular local ring if and only if \(R\) is a commutative semihereditary local ring. Lugansk National Taras Shevchenko University 2018-10-20 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/503 Algebra and Discrete Mathematics; Vol 26, No 1 (2018) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/503/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/503/232 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Bezout ring Hermite ring elementary divisor ring semihereditary ring regular ring neat ring clean ring stable range 1 13F99 06F20 Zabavsky, Bohdan Volodymyrovych Type conditions of stable range for identification of qualitative generalized classes of rings |
| 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 |
| topic | Bezout ring Hermite ring elementary divisor ring semihereditary ring regular ring neat ring clean ring stable range 1 13F99 06F20 |
| topic_facet | Bezout ring Hermite ring elementary divisor ring semihereditary ring regular ring neat ring clean ring stable range 1 13F99 06F20 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/503 |
| work_keys_str_mv | AT zabavskybohdanvolodymyrovych typeconditionsofstablerangeforidentificationofqualitativegeneralizedclassesofrings |