Writing units of integral group rings of finite abelian groups as a product of Bass units. (Q2862540)

A very well-known result of Bass and Milnor says that if \(G\) is a finite abelian group, then the Bass units of the integral group ring \(\mathbb ZG\) generate a subgroup of finite index in its unit group \(\mathcal U(\mathbb ZG)\). This means that if \(u\in\mathcal U(\mathbb ZG)\), then a power of \(u\) is a product of Bass units.NEWLINENEWLINE In the paper under review the authors provide a new and constructive proof of the Bass-Milnor Theorem. As a result it provides an algorithm that for a cyclotomic unit \(\eta\) as input, returns \(m\) and an expression of \(\eta^m\) as a product of Bass units. This covers Section 2 of the manuscript. Furthermore a concrete basis formed by Bass units for a free abelian subgroup of finite index in \(\mathcal U(\mathbb ZG)\) is given in Section 3. Finally in the last Section a set of multiplicatively independent units generating a subgroup of finite index in \(\mathcal U(\mathbb ZG)\) is obtained in the case in which \(G\) is either an elementary abelian \(p\)-group (Proposition 4.1) or a cyclic \(p\)-group (Proposition 4.2).