Geodesic
Zero Divisors and Idempotents in Group Rings
Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 596-602

Voir la notice de l'article provenant de la source Cambridge University Press

We consider the following problem: If KG is the group ring of a torsion free group over a field K,show that KG has no divisors of zero. At characteristic zero, major progress was made by Brown [2], who solved the problem for G abelian-by-finite, and then by Farkas and Snider [4], who considered Gpolycyclic-by-finite. Here we present a solution at nonzero characteristic for polycyclic-by-finite groups. We also show that if Khas characteristic p > 0 and G is polycyclic-by-finite with only p-torsion, then KG has no idempotents other than 0 or 1. Finally we show that if R is a commutative ring of nonzero characteristic without nontrivial idempotents and G is polycyclic-by-finite such that no element different from 1 in G has order invertible in R, then RG has no nontrivial idempotents. This is proved at characteristic zero in [3]. 
Cliff, Gerald H. Zero Divisors and Idempotents in Group Rings. Canadian journal of mathematics, Tome 32 (1980) no. 3, pp. 596-602. doi: 10.4153/CJM-1980-046-3
@article{10_4153_CJM_1980_046_3,
     author = {Cliff, Gerald H.},
     title = {Zero {Divisors} and {Idempotents} in {Group} {Rings}},
     journal = {Canadian journal of mathematics},
     pages = {596--602},
     year = {1980},
     volume = {32},
     number = {3},
     doi = {10.4153/CJM-1980-046-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-046-3/}
}
TY  - JOUR
AU  - Cliff, Gerald H.
TI  - Zero Divisors and Idempotents in Group Rings
JO  - Canadian journal of mathematics
PY  - 1980
SP  - 596
EP  - 602
VL  - 32
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-046-3/
DO  - 10.4153/CJM-1980-046-3
ID  - 10_4153_CJM_1980_046_3
ER  - 
%0 Journal Article
%A Cliff, Gerald H.
%T Zero Divisors and Idempotents in Group Rings
%J Canadian journal of mathematics
%D 1980
%P 596-602
%V 32
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-046-3/
%R 10.4153/CJM-1980-046-3
%F 10_4153_CJM_1980_046_3

[1] 1. Bass, H., Euler characteristics and characters of discrete groups, Inv. Math. 35 (1976), 155–196. Google Scholar

[2] 2. Brown, K. A., On zero divisors in group rings, Bull. London Math. Soc. 8 (1976), 251–256. Google Scholar

[3] 3. Cliff, G. and Sehgal, S., On the trace of an idempotent in a group ring, Proc. Amer. Math. Soc. 62 (1977), 11–14. Google Scholar

[4] 4. Farkas, D. and Snider, R., Ko and Noetherian group rings, J. Algebr. 42 (1976), 192–198. Google Scholar

[5] 5. Formanek, E., Idempotents in Noetherian group rings, Can. J. Math. 15 (1973), 366–369. Google Scholar

[6] 6. Passman, D., The algebraic structure of group rings (Wiley-Interscience, New York, 1977). Google Scholar

[7] 7. Sehgal, S., Topics in group rings (Marcel Dekker, New York, 1978). Google Scholar

[8] 8. Serre, J.-P., Corps locaux (Hermann, Paris, 1962). Google Scholar

[9] 9. Zariski, O. and Samuel, P., Commutative algebra (Vol. I) (Van Nostrand, Princeton, 1958). Google Scholar

Cité par Sources :