On group rings of nilpotent groups
Let F be a field and G a torsion-free non-abelian nilpotent group. It is well known that G contains a subgroup \(H=<a,b;\quad [a,b]=c,\quad [a,c]=[b,c]=1>.\) The group algebras F[H] and F[G] are Ore domains and their skew fields of fractions are denoted by F(H) and F(G), respectively. It is proved that the elements \((1-a)^{-1}\) and \((1- a)^{-1}(1-b)^{-1}\) generate a free subalgebra of F(H) of rank 2. It follows that F(H), and hence F(G), contains a free subalgebra of any countable rank.
torsion-free non-abelian nilpotent group
scientific article; zbMATH DE number 3886073