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Title: | Type conditions of stable range for identification of qualitative generalized classes of rings |
Authors: | Zabavsky, B. |
Keywords: | Bezout ring Hermite ring elementary divisor ring semihereditary ring regular ring neat ring clean ring stable range 1. |
Issue Date: | 2018 |
Publisher: | ДЗ "Луганський національний університет імені Тараса Шевченка" |
Series/Report no.: | Математичні науки; |
Abstract: | 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. |
Description: | Zabavsky B. Type conditions of stable range for identification of qualitative generalized classes of rings / B. Zabavsky // Algebra and Discrete Mathematics. - 2018. - Vol. 26. - Number 1. - Рp. 144- 152 |
URI: | http://hdl.handle.net/123456789/4541 |
Appears in Collections: | Algebra and Discrete Mathematics. - № 1 (26). - 2018 |
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503-3654-1-PB (1).pdf | 317.08 kB | Adobe PDF | View/Open |
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