Idempotents in exchange rings (Q2778263)

An associative ring \(R\) is called exchange 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, there exist submodules \(A_i'\subseteq A_i\) such that \(A=M\oplus(\bigoplus_{i\in I}A_i')\). On the other hand we say that \(R\) has stable range one provided that \(aR+bR=R\) implies that \(a+by\in U(R)\) for some \(y\in R\). The author shows that an exchange ring \(R\) has stable range one if and only if \(eR\cong fR\) with idempotents \(e,f\in R\) implies \(e=ufu^{-1}\) for some \(u\in U(R)\).