On Lifting of Stable Range One Elements
J. Korean Math. Soc.
Published online September 24, 2019
Meltem Altun Ozarslan and Ayse Cigdem Ozcan
Hacettepe University, Department of Mathematics, Ankara, Turkey
Abstract : Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules.
The newly appeared element-wise setting for the simplest case of stable range one is tempting to study
the lifting property modulo ideals. We study the lifting of elements having (idempotent) stable range
one from a quotient of a ring $R$ modulo a two-sided ideal $I$ by providing several examples and
investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring $R$ is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if $R$ is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.
Keywords : Stable range one, idempotent stable range one, unit-regular, lifting of units
MSC numbers : 16E50, 16D25, 16U99
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