Direct sum decompositions of torsion-free Abelian groups of finite rank (Q2738744)

A torsion-free Abelian group \(A\) of finite rank has the unique decomposition of subgroups property (UDS), if finite direct sums of groups quasi-isomorphic to \(A\) have unique direct sum decompositions into indecomposable summands.NEWLINENEWLINENEWLINEThe author characterizes some classes of strongly indecomposable groups with the UDS property in terms of properties of their endomorphism rings. For example, if \(A\) is a strongly indecomposable torsion-free group of finite rank, then the following three properties are equivalent: (a) \(A\) has the UDS property, (b) Each torsion-free finite rank group quasi-isomorphic to \(A\) has the one-sided UDS property, (c) There is at most one prime \(p\) with \(p\text{-rank}(A)>1\) and either \(S_A/pS_A\) is a local ring or else \(S_A\) has exactly two maximal right ideals \(M_1\) and \(M_2\) containing \(p\) such that \(M_1\) is principal, \(G_A/M_1G_A\) is a cyclic group, and \(M_1/pS_A\) has the unique maximal condition in \(S_A/pS_A\). This is then related to the UDI (rings' unique decomposition into ideals) property: If \(A\) is a finite-rank torsion-free group and \(\mathbb{Q}\text{End }A\) is a field, such that \(A\) has the UDS property, then \(\text{End }A\) has the UDI property.NEWLINENEWLINENEWLINESome applications involve the Krull-Schmidt groups (torsion-free groups of finite rank with the property that direct sum decompositions into indecomposable groups are unique up to isomorphism and order): For an odd prime \(p\), there is a \(p\)-local torsion-free group \(A\) of rank 10 that is not a Krull-Schmidt group.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].