Stable range conditions for abelian and duo rings

A. A. Dmytruk; A. I. Gatalevych; M. I. Kuchma

doi:10.30970/ms.57.1.92-97

A. A. Dmytruk Ivan Franko National University of L'viv https://orcid.org/0000-0002-2393-0193
A. I. Gatalevych Ivan Franko National University of L'viv
M. I. Kuchma Lviv Polytechnic National University

DOI: https://doi.org/10.30970/ms.57.1.92-97

Keywords: duo ring; abelian ring; stable range; elementary divisor ring

Abstract

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).

Author Biographies

A. A. Dmytruk, Ivan Franko National University of L'viv

Ivan Franko National University of L'viv

A. I. Gatalevych, Ivan Franko National University of L'viv

Department of Higher Mathematics,

Head of Department

M. I. Kuchma, Lviv Polytechnic National University

Department of Higher Mathematics,

Assoc. Prof.

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