The Krull-Schmidt property for ideals and modules over integral domains. (Q1415005)

Let \(R\) be a commutative integral domain and \(\mathcal{C}\) a class of \(R\)-modules. The Krull-Schmidt property holds for \(\mathcal{C}\) if, whenever \[ G_1\oplus G_2\oplus\cdots\oplus G_n\cong H_1\oplus H_2\oplus\cdots\oplus H_m \] for \(G_i, H_j\in\mathcal {C}\), then \(n=m\) and, after reindexing, \(G_i\cong H_i\) for all \(i\leq n\). If, instead of \(G_i\cong H_i\), we require only that \(k>0\) exists such that \(G_i^{(k)}\cong H_i^{(k)}\) for all \(i\), then we say the weak Krull-Schmidt property holds for \(\mathcal{C}\) (we write \(G^{(k)}\) for a direct sum of \(k\) copies of a module \(G\).) In the paper under review the authors examine when versions of the Krull-Schmidt property hold for the following: (1) direct sums of ideals of integral domains, (2) direct sums of indecomposable submodules of finitely generated free modules, (3) direct sums of rank one torsion-free modules.