Clear rings and clear elements

B. V. Zabavsky; O. V. Domsha; O. M.  Romaniv

doi:10.30970/ms.55.1.3-9

B. V. Zabavsky Ivan Franko National University, Lviv, Ukraine
O. V. Domsha Lviv Regional Institute for Public Administration of the National Academy for Public Administration under the President of Ukraine, Lviv, Ukraine
O. M. Romaniv Ivan Franko National University of Lviv

DOI: https://doi.org/10.30970/ms.55.1.3-9

Keywords: Bezout domain, clean element, unit-regular element, full matrix, elementary divisor ring, clear element, clear ring

Abstract

An element of a ring $R$ is called clear if it is a sum of a unit-regular element and a unit. An associative ring is clear if each of its elements is clear.
In this paper we defined clear rings and extended many results to a wider class. Finally, we proved that a commutative Bezout domain is an elementary divisor ring if and only if every full $2\times 2$ matrix over it is nontrivially clear.

Author Biographies

B. V. Zabavsky, Ivan Franko National University, Lviv, Ukraine

Professor
Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

O. V. Domsha, Lviv Regional Institute for Public Administration of the National Academy for Public Administration under the President of Ukraine, Lviv, Ukraine

Lviv Regional Institute for Public Administration of the National Academy for Public Administration under the President of Ukraine, Lviv, Ukraine

O. M. Romaniv, Ivan Franko National University of Lviv

Associate Professor,
Department of Algebra and Logic,
Faculty of Mechanics and Mathematics,
Ivan Franko National University of Lviv

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