Invariant ideals and polynomial forms

Author:
D. S. Passman

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3379-3408

MSC (2000):
Primary 16S34; Secondary 20F50, 20G05

Published electronically:
April 1, 2002

MathSciNet review:
1897404

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the group algebra of an infinite locally finite group . In recent years, the lattice of ideals of has been extensively studied under the assumption that is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra stable under the action of an appropriate automorphism group of . Specifically, in this paper, we let be a quasi-simple group of Lie type defined over an infinite locally finite field , and we let be a finite-dimensional vector space over a field of the same characteristic . If acts nontrivially on by way of the homomorphism , and if has no proper -stable subgroups, then we show that the augmentation ideal is the unique proper -stable ideal of when . The proof of this result requires, among other things, that we study characteristic division rings , certain multiplicative subgroups of , and the action of on the group algebra , where is the additive group . In particular, properties of the quasi-simple group come into play only in the final section of this paper.

**[Be]**V. V. Belyaev,*Locally finite Chevalley groups*, Studies in group theory, Akad. Nauk SSSR, Ural. Nauchn. Tsentr, Sverdlovsk, 1984, pp. 39–50, 150 (Russian). MR**818993****[BHPS]**K. Bonvallet, B. Hartley, D. S. Passman, and M. K. Smith,*Group rings with simple augmentation ideals*, Proc. Amer. Math. Soc.**56**(1976), 79–82. MR**0399145**, 10.1090/S0002-9939-1976-0399145-0**[BT]**Armand Borel and Jacques Tits,*Homomorphismes “abstraits” de groupes algébriques simples*, Ann. of Math. (2)**97**(1973), 499–571 (French). MR**0316587****[Bo]**A. V. Borovik,*Periodic linear groups of odd characteristic*, Dokl. Akad. Nauk SSSR**266**(1982), no. 6, 1289–1291 (Russian). MR**681626****[BE]**C. J. B. Brookes and D. M. Evans,*Augmentation modules for affine groups*, Math. Proc. Cambridge Philos. Soc.**130**(2001), 287-294. CMP**2001:06****[C]**Roger W. Carter,*Simple groups of Lie type*, John Wiley & Sons, London-New York-Sydney, 1972. Pure and Applied Mathematics, Vol. 28. MR**0407163****[HS]**B. Hartley and G. Shute,*Monomorphisms and direct limits of finite groups of Lie type*, Quart. J. Math. Oxford Ser. (2)**35**(1984), no. 137, 49–71. MR**734665**, 10.1093/qmath/35.1.49**[HZ1]**Jan Ambrosiewicz,*On conjugacy classes in linear groups*, Demonstratio Math.**26**(1993), no. 2, 359–362. MR**1240225****[HZ2]**B. Hartley and A. E. Zalesskiĭ,*Confined subgroups of simple locally finite groups and ideals of their group rings*, J. London Math. Soc. (2)**55**(1997), no. 2, 210–230. MR**1438625**, 10.1112/S0024610796004759**[J]**Gerald J. Janusz,*Algebraic number fields*, 2nd ed., Graduate Studies in Mathematics, vol. 7, American Mathematical Society, Providence, RI, 1996. MR**1362545****[LP]**F. Leinen and O. Puglisi,*Ideals in group algebras of simple locally finite groups of -type*, to appear.**[OPZ]**J. M. Osterburg, D. S. Passman, and A. E. Zalesski,*Invariant ideals of abelian group algebras under the multiplicative action of a field, II*, Proc. Amer. Math. Soc., to appear.**[P]**Donald S. Passman,*The algebraic structure of group rings*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR**470211****[PZ]**D. S. Passman and A. E. Zalesski,*Invariant ideals of abelian group algebras and representations of groups of Lie type*. Trans. Amer. Math. Soc.**353**(2001), 2971-2982.**[RS]**J. E. Roseblade and P. F. Smith,*A note on hypercentral group rings*, J. London Math. Soc. (2)**13**(1976), no. 1, 183–190. MR**0399146****[S]**Gary M. Seitz,*Abstract homomorphisms of algebraic groups*, J. London Math. Soc. (2)**56**(1997), no. 1, 104–124. MR**1462829**, 10.1112/S0024610797005176**[St]**Robert Steinberg,*Lectures on Chevalley groups*, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR**0466335****[T]**Simon Thomas,*The classification of the simple periodic linear groups*, Arch. Math. (Basel)**41**(1983), no. 2, 103–116. MR**719412**, 10.1007/BF01196865**[Z1]**A. E. Zalesski,*Intersection theorems in group rings*(in Russian), No. 395-74, VINITI, Moscow, 1974.**[Z2]**A. E. Zalesskiĭ,*Group rings of inductive limits of alternating groups*, Algebra i Analiz**2**(1990), no. 6, 132–149 (Russian); English transl., Leningrad Math. J.**2**(1991), no. 6, 1287–1303. MR**1092530****[Z3]**A. E. Zalesskiĭ,*A simplicity condition for the fundamental ideal of the modular group algebra of a locally finite group*, Ukrain. Mat. Zh.**43**(1991), no. 7-8, 1088–1091 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J.**43**(1991), no. 7-8, 1021–1024 (1992). MR**1148874**, 10.1007/BF01058712**[Z4]**A. E. Zalesskiĭ,*Group rings of simple locally finite groups*, Finite and locally finite groups (Istanbul, 1994) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 471, Kluwer Acad. Publ., Dordrecht, 1995, pp. 219–246. MR**1362812**, 10.1007/978-94-011-0329-9_9

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
16S34,
20F50,
20G05

Retrieve articles in all journals with MSC (2000): 16S34, 20F50, 20G05

Additional Information

**D. S. Passman**

Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Email:
passman@math.wisc.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03006-4

Received by editor(s):
November 16, 2001

Published electronically:
April 1, 2002

Additional Notes:
Research supported in part by NSF Grant DMS-9820271.

Dedicated:
Dedicated to Idun Reiten on the occasion of her $60$th birthday

Article copyright:
© Copyright 2002
American Mathematical Society