Weakly exchange rings whose units are sums of two idempotents

Abstract

We prove that if every element in the unit group U(R) of a weakly exchange ring R is a sum of two idempotents of R, then every element in the center C(R) of R is a sum of two central idempotents of R. This somewhat enlarges results due to Ko¸san Ying Zhou published in Can. Math. Bull. (2016) as well as due to Karimi Ko¸san Zhou published in Contemp. Math. (2018). Moreover, we show that each nilpotent of order not exceeding 2 in a von Neumann regular ring is a difference of two special (left-right symmetric) idempotents. This somewhat refines a recent result by O’Meara stated in a still unpublished preprint (2018).