On the splitting group basis problem for Abelian group rings (Q1183356)

Let \(K\) be a field and \(G\) an Abelian group with torsion subgroup \(T\) having no element of order \(char(K)\). To the field \(K\) and the group \(T\) the authors associate a subgroup \(V\) of \(T\) containing \(\bigcap_{n \in \mathbb{N}}T^ n\) and an Abelian torsion group \(T'\) containing \(T/V\). Let \(G'\) be the amalgamated product of \(G/V\) and \(T'\) over \(T/V\). It is shown that (1) If \(T'\) is a direct summand of \(G'\), then \(V(KT)\) is a direct summand of \(V(KG)\) \((V(KT)\) is the group of units of \(KT\) with augmentation 1). (2) The converse also holds in each of the following cases: (a) For every \(p\), \(T_ p/V_ p\) is a direct sum of cyclic \(p\)- groups or is \(p\)-torsion complete. (b) \(G/T\) is countable. Earlier results in this direction were obtained by May (1971), Berman and Mollov (1975) and Berman and Bogdan (1977).