Group rings in which the group of units is hyperbolic. (Q2903539)

(Semi-) group rings with hyperbolic groups of units have been studied in several research papers recently, such as \textit{E. Iwaki, E. Jespers, S. O. Juriaans}, and \textit{A. C. Souza Filho} [J. Algebra Appl. 9, No. 6, 871-876 (2010; Zbl 1209.16032)], and \textit{E. Iwaki} and \textit{S. O. Juriaans} [Commun. Algebra 36, No. 4, 1336-1345 (2008; Zbl 1149.16028)]. The author gives here a more detailed description.NEWLINENEWLINE Let \(K\) be a commutative ring with unity, \(G\) a group with nontrivial torsion part. If \(K\) is of zero characteristic and \(U(KG)\) is hyperbolic then either \(G\) is cyclic of order 5, 8 or 12; or finite Abelian of exponent 2, 3, 4 or 6; or a Hamiltonian 2-group; or the semidirect product of a cyclic group of order either 3 or 4 by a cyclic group of order either 2 or 4 acting by inversion; or the semidirect product of either a finite Hamiltonian 2-group or a finite Abelian group of exponent 2, 3, 4 or 6 by an infinite cyclic group either inverting all torsion elements or leaving them fixed. If \(K\) is a field of positive characteristic then \(U(KG)\) is hyperbolic if and only if both \(K\) and \(G\) are finite.