Localized group rings, the invariant basis property and Euler characteristics

Author:
K. R. Goodearl

Journal:
Proc. Amer. Math. Soc. **91** (1984), 523-528

MSC:
Primary 16A27

MathSciNet review:
746081

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The technique of embedding a complex group algebra into the von Neumann regular ring associated with the corresponding group algebra is exploited to prove that certain localizations of a group ring possess the invariant basis property. From this it follows, using a method of S. Rosset, that certain -modules have zero Euler characteristic. The assumptions are that is a commutative integral domain of characteristic zero, and that has a nontrivial, torsion-free, abelian normal subgroup . The main result of the paper is that the localization of obtained by inverting all elements of the form , where is a nonzero element of and is a nontrivial element of , has the invariant basis property; more generally, this localization and all its matrix rings are directly finite. (M. Smith has extended the methods of this paper to cover the localization of obtained by inverting all nonzero elements of .) Given a -module which has a finite free resolution, such that is finitely generated over , it is proved that the Euler characteristic of is zero. This verifies an unpublished result of Rosset.

**[1]**S. K. Berberian,*The maximal ring of quotients of a finite von Neumann algebra*, Rocky Mountain J. Math.**12**(1982), no. 1, 149–164. MR**649748**, 10.1216/RMJ-1982-12-1-149**[2]**Steven A. Gaal,*Linear analysis and representation theory*, Springer-Verlag, New York-Heidelberg, 1973. Die Grundlehren der mathematischen Wissenschaften, Band 198. MR**0447465****[3]**K. R. Goodearl,*Ring theory*, Marcel Dekker, Inc., New York-Basel, 1976. Nonsingular rings and modules; Pure and Applied Mathematics, No. 33. MR**0429962****[4]**-,*Von Neumann regular rings*, Pitman, London, 1979.**[5]**Irving Kaplansky,*Fields and rings*, The University of Chicago Press, Chicago, Ill.-London, 1969. MR**0269449****[6]**Irving Kaplansky,*Commutative rings*, Allyn and Bacon, Inc., Boston, Mass., 1970. MR**0254021****[7]**George W. Mackey,*The theory of unitary group representations*, University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955; Chicago Lectures in Mathematics. MR**0396826****[8]**F. I. Mautner,*Unitary representations of locally compact groups. I*, Ann. of Math. (2)**51**(1950), 1–25. MR**0032650****[9]**F. J. Murray and J. Von Neumann,*On rings of operators*, Ann. of Math. (2)**37**(1936), no. 1, 116–229. MR**1503275**, 10.2307/1968693**[10]**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****[11]**J. T. Schwartz,*𝑊*-algebras*, Gordon and Breach Science Publishers, New York-London-Paris, 1967. MR**0232221****[12]**Martha K. Smith,*Central zero divisors in group algebras*, Proc. Amer. Math. Soc.**91**(1984), no. 4, 529–531. MR**746082**, 10.1090/S0002-9939-1984-0746082-X**[13]**John Stallings,*Centerless groups—an algebraic formulation of Gottlieb’s theorem*, Topology**4**(1965), 129–134. MR**0202807****[14]**Bo Stenström,*Rings of quotients*, Springer-Verlag, New York-Heidelberg, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217; An introduction to methods of ring theory. MR**0389953****[15]**D. M. Topping,*Lectures on Von Neumann algebras*, Van Nostrand Reinhold, New York, 1971.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
16A27

Retrieve articles in all journals with MSC: 16A27

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1984-0746081-8

Keywords:
Group ring,
invariant basis property,
direct finiteness,
Euler characteristic

Article copyright:
© Copyright 1984
American Mathematical Society