Abstract
For a commutative ring R with identity and an arbitrary group G, let RG denote the group ring of G over R and U(RG) its group of units. It is of interest, see the survey by Dennis (1977), to determine the necessary and sufficient conditions on R and G in order that U(RG) has a specific group-theoretic property, e.g., solvability, nilpotence, etc.. Considerable work has been done, by various authors, on these questions over the last thirty years or so. Besides solvability and nilpotence, other group-theoretic problems for the unit groups of group rings like the characterization of residual nilpotence, residual solvability, being an FC-group, torsion elements forming a subgroup, as well as the behaviour of the upper central series of the unit groups have also been studied. S.K. Sehgal’s book (1978) covers the main results obtained in this direction up to 1977 while the article by C. Polcino Milies (1981) gives some later developments (see also the books of Sehgal, 1989 and Karpilovsky, 1989). In this article our main aim is to survey the more recent developments. In § 1 we review the case when R is a field and in §2 the case of the integral group ring is considered.
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Bhandari, A.K., Passi, I.B.S. (1999). Unit Groups of Group Rings. In: Passi, I.B.S. (eds) Algebra. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9996-3_2
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