Elements in exchange rings with related comparability (Q1578345)

An associative unital ring \(R\) is said to satisfy related comparability provided that for any idempotents \(e\) and \(f\) in \(R\) such that \(e=1+ab\) and \(f=1+ba\) for some \(a,b\in R\), there exists a central idempotent \(u\) in \(R\) such that \(ueR\) is isomorphic to a direct summand of \(ufR\) and \((1-u)fR\) is isomorphic to a direct summand of \((1-u)eR\). In addition, \(R\) is said to be an exchange ring if for every right \(R\)-module \(A\) and any two decompositions \(A=M\oplus N=\bigoplus_{i\in I}A_i\), where \(M_R\cong R\) and the index set \(I\) is finite, then there exist submodules \(A_i'\subseteq A_i\) such that \(A=M\oplus(\bigoplus_{i\in I}A_i')\). The author shows that if \(R\) is an exchange ring, then the following are equivalent: (1) \(R\) satisfies related comparability; (2) Given \(a,b,d\in R\) with \(aR+bR=dR\), there exists a related unit \(u\in R\) such that \(a+bt=du\); (3) Given \(a,b\in R\) with \(aR=bR\), there exists a related unit \(u\in R\) such that \(a=bu\).