Journal of the
Korean Mathematical Society JKMS

In this paper, we introduce a new class of rings, $SB$-rings. We establish various properties of this concept. These shows that, in several respects, $SB$-rings behave like rings satisfying unit $1$-stable range. We will give necessary and sufficient conditions under which a semilocal ring is a $SB$-ring. Furthermore, we extend these results to exchange rings with all primitive factors artinian. For such rings, we observe that the concept of the $SB$-ring coincides with Goodearl--Menal condition. These also generalize the results of Huh et al., Yu and the author on rings generated by their units.

Keywords: $SB$-ring, semilocal ring, exchange ring

MSC numbers: 16E50, 19U99, 15A33