ON LIFTING OF STABLE RANGE ONE ELEMENTS

. Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly ap- peared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lift- ing 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.