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Group Rings over Z(p) with FC Unit Groups

Published online by Cambridge University Press:  20 November 2018

H. Merklen
Affiliation:
Universidade de São Paulo, São Paulo, Brasil
C. Polcino Milies
Affiliation:
Universidade de São Paulo, São Paulo, Brasil
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Let RG denote the group ring of a group G over a commutative ring R with unity. We recall that a group is said to be an FC-group if all its conjugacy classes are finite.

In [6], S. K. Sehgal and H. Zassenhaus gave necessary and sufficient conditions for U(RG) to be an FC-group when R is either Z, the ring of rational integers, or a field of characteristic 0.

One of the authors considered this problem for group rings over infinite fields of characteristic p ≠ 2 in [5] and G. Cliffs and S. K. Sehgal [1] completed the study for arbitrary fields. Also, group rings of finite groups over commutative rings containing Z(p), a localization of Z over a prime ideal (p) were studied in [4].

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Cliff, G. and Sehgal, S. K., Group rings whose units form an FC-group, to appear.Google Scholar
2. Coleman, D. B., Idempotents in group rings, Proc. Amer. Math. Soc. 17 (1966), 962.Google Scholar
3. Fuchs, L., Infinite abelian groups, Vol. I (Academic Press, New York, 1970).Google Scholar
4. Parmenter, M. M. and Polcino Milies, C. , Group rings whose units form a nilpotent or FC-group, Proc. Amerc. Math. Soc, to appear.CrossRefGoogle Scholar
5. Milies, C. Polcino, Group rings whose units form an FC-group, Archiv der Math., to appear.Google Scholar
6. Sehgal, S. K. and Zassenhaus, H. J., Group rings whose units form an FC-group, Math. Z. 153 (1977), 2935.Google Scholar
7. Zassenhaus, H. J., On the torsion units of finite group rings, in Studies in Mathematics (in honor of A. Almeida Costa), Instituto de Alta Cultura, Lisboa (1974).Google Scholar
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