In [J. Algebra 230, No. 2, 608-655 (2000; Zbl 0963.16008)], P. Ara, G. K. Pedersen and F. Perera introduced the class of \(QB\)-rings, to give an infinite analog of rings with stable range \(1\). The present author generalized this concept to \(QB_\infty\)-rings [in Commun. Algebra 34, No. 6, 2057-2068 (2006; Zbl 1096.19001)] as follows. First, an element \(u\) in a ring \(R\) is defined to be pseudo invertible in case there exist \(v,w\in R\) such that the ideal \(R(1-uv)R(1-wu)R\) is nilpotent. Then, \(R\) is \(QB_\infty\) provided that whenever \(a,b\in R\) and \(aR+bR=R\), there is some \(y\in R\) such that \(a+by\) is pseudo invertible. The author established various necessary and sufficient conditions for an exchange ring to be \(QB_\infty\) [in Algebra Colloq. 14, No. 4, 613-623 (2007; Zbl 1143.16007)], and he proved some cancellation and diagonalization results for \(QB_\infty\) exchange rings there.
Here, the author gives further characterizations of \(QB_\infty\) exchange rings, such as the following: An exchange ring \(R\) is \(QB_\infty\) if and only if every (von Neumann) regular element \(x\in R\) is pseudo unit-regular (meaning that \(xux=x\) for some pseudo invertible element \(u\in R\)); if and only if every regular element of \(R\) is a product of an idempotent and a pseudo invertible element; if and only if whenever \(x=xyx\) in \(R\), there is a pseudo invertible element \(u\in R\) such that \(x=xyu=uyx\). Many of these results are \(QB_\infty\) analogs of characterizations of exchange rings with weak stable range \(1\) given by J. Wei [in Vietnam J. Math. 32, No. 4, 441-449 (2004; Zbl 1085.16008) and Commun. Algebra 33, No. 6, 1937-1946 (2005; Zbl 1093.16003)].