Journal of the
Korean Mathematical Society JKMS

\noindent 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

Supported by: This work was supported by Hacettepe University Scientific Research Projects Coordination Unit (Project No. FDK-2018-16894). Also, the first author would like to thank The Scientific and Technological Research Council of Turkey (TUBITAK) for financial support.