On Utumi rings and continuous regular Baer rings

Abstract

In 1963, Utumi determined under what condition the maximal one-sided ring of quotients of a ring is, in fact, two-sided. That is, he showed if a ring is nonsingular and cononsingular then its maximal right and left rings of quotients coincide. In this paper, we provide more properties of an Utumi ring. A ring is called (right) Utumi if it is both (right) nonsingular and (right) cononsingular. We show that every right Utumi ring is left nonsingular. Moreover, we prove that, for any one-sided nonsingular ring R R , its maximal right and left rings of quotients coincide if and only if R R is a cononsingular ring. Thus, for a right nonsingular R R , which is not left nonsingular, the maximal right and left rings of quotients of R R are different. We also prove that every left cononsingular, right extending ring is directly finite. In particular, every left cononsingular, right nonsingular, right continuous ring is a unit-regular Baer ring. Furthermore, we characterize right and left continuous regular rings in terms of Utumi rings. Examples which delineate the concepts and results are provided.