Stable range one for exchange rings (Q1346823)

Let \(R\) be an associative ring with 1. \(R\) is called an exchange ring if for any two decompositions of a right \(R\)-module \(A\) of the form \(A\cong R\oplus N\cong\bigoplus_{i\in I}A_i\) there exist submodules \(A_i'\) of \(A_i\) such that \(A \cong R\oplus(\bigoplus_{i\in I}A_i')\). The author shows that exchange rings have stable range 1 under any of the following additional hypotheses: i) the primitive factors of \(R\) are Artinian; ii) the index of nilpotency of \(R\) is bounded; iii) the idempotents of \(R\) are central.