Exchange rings satisfying the related comparability. (English) Zbl 1030.16005

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.


16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16E20 Grothendieck groups, \(K\)-theory, etc.
16U60 Units, groups of units (associative rings and algebras)
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