On modules with finite exchange property (Q1597740)

Let \(M\) be a right module over an associative ring \(R\) with identity element. Recall that \(M\) has the finite exchange property if for any finite index set \(I\) and any module decomposition \(N=M\oplus K=\bigoplus_{i\in I}D_i\), there are submodules \(D_i'\) of \(D_i\) such that \(N=M\oplus(\bigoplus_{i\in I}D_i')\). It is shown that if a module \(M=\bigoplus_{i\in I}M_i\) is a direct sum of indecomposable injective modules and has the finite exchange property, then for each direct summand \(N\) of \(M\) there exists a subset \(J\) of \(I\) such that \(N=\bigoplus_{j\in J}M_j\).