Exchange rings satisfying the related comparability
An exchange ring \(R\) is said to satisfy the related comparability, provided that for any idempotents \(e,f\in R\) with \(e=1+ab\) and \(f=1+ba\) for some \(a\in(1-e)R(1-f)\) and \(b\in(1-f)R(1-e)\), there exists a central idempotent \(u\in R\) such that \(ueR\lesssim^\oplus ufR\) and \((1-u)fR\lesssim^\oplus(1-u)eR\). It is shown that an exchange ring \(R\) satisfies the related comparability if and only if for any regular \(x\in R\), there exist a related unit \(w\in R\) and a group \(G\) in \(R\) such that \(wx\in G\). In addition, we investigate the uniqueness of related units for generalized inverses in exchange rings.
scientific article; zbMATH DE number 1943352