Abstract
According to [Comm. Algebra 46 (2018), no. 2, 870-886], a ring R is called right Utumi-ring (U-ring) if, whenever A and B are right ideals of R with A congruent to B and A n B = 0, there exist idempotents e, f and g of R such that A subset of(ess) eR, B subset of(ess) fR and eR circle plus fR = gR. The class of U-rings is a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant rings. In this paper, we prove that every right U-ring that is totally non-abelian (i.e. every non-zero right ideal of R contains a non-central idempotent) is generated as a ring by its idempotents. Moreover, under these assumptions R turns out to be regular and right self-injective. This result extends the work of Utumi who proved that every totally non-abelian regular right self-injective ring is generated as a ring by its idempotents.