Comparability and \(s\)-comparability of modules over exchange rings (Q5946547)

A ring \(R\) is called an exchange ring if for every right \(R\)-module \(A\) and any two decomposition \(A=M\oplus N=\bigoplus_{i\in I}A_i\), where \(M_R\cong R_R\) and the index set \(I\) is finite, there are submodules \(B_i\) of \(A_i\) such that \(A=M\oplus(\bigoplus_{i\in I}B_i)\). It is shown that the subdirect product of an exchange ring with stable range one and an exchange ring with comparability is also an exchange ring satisfying comparability. Also \(s\)-comparability over exchange rings is studied.