Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On fully invariant ideals
of the free group algebra


Authors: A. N. Krasil'nikov and Samuel M. Vovsi
Journal: Proc. Amer. Math. Soc. 124 (1996), 2613-2618
MSC (1991): Primary 16S34, 20C07
DOI: https://doi.org/10.1090/S0002-9939-96-03502-2
MathSciNet review: 1343706
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $RF$ be the group algebra of a free noncyclic group $F$ over an integral domain $R$. It is proved that if $R$ is not a field, then there exists a fully invariant ideal $I$ of $RF$ such $RF/I$ is torsion-free but not projective as an $R$-module. In other words, there exists a pure nonprojective variety of group representations over $R$.


References [Enhancements On Off] (What's this?)

  • [K] A. N. Krasil'nikov, On additive groups of free objects in varieties of integral group representations, Commun. Algebra 23 (1995), 1231--1238. MR 95k:16036
  • [M] W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann. 111 (1935), 259--280.
  • [N] H. Neumann, Varieties of Groups, Springer, Berlin, 1967. MR 35:6734
  • [P] B. I. Plotkin, Varieties of group representations, Uspekhi Mat. Nauk 32 (1977), no. 5, 3--68; English transl, Russian Math. Surveys 32 (1977), no. 5, 1--72. MR 57:9849
  • [PV] B. I. Plotkin and S. M. Vovsi, Varieties of Group Representations: General Theory, Connections and Applications, Zinatne, Riga, 1983 (Russian). MR 86e:20001
  • [S] A. Storozhev, On abelian subgroups of relatively free groups, Commun. Algebra 22 (1994), 2677--2701. MR 95d:20066
  • [V1] S. M. Vovsi, Topics in Varieties of Group Representations, London Math. Soc. Lecture Notes, vol. 163, Cambr. Univ. Press, Cambridge, 1991. MR 93i:20015
  • [V2] S. M. Vovsi, On the semigroups of fully invariant ideals of the free group algebra and the free associative algebra, Proc. Amer. Math. Soc. 119 (1993), 1029--1037. MR 94a:16050
  • [ZhS] K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, and A. I. Shirshov, Rings That Are Nearly Associative, Nauka, Moscow, 1978 (Russian). MR 80h:17002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 16S34, 20C07

Retrieve articles in all journals with MSC (1991): 16S34, 20C07


Additional Information

A. N. Krasil'nikov
Affiliation: Department of Algebra, Moscow State Pedagogical University, Moscow 119882, Russia
Email: krasilnikov.algebra@mpgu.msk.su

Samuel M. Vovsi
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Email: vovsi@math.ias.edu, vovsi@math.rutgers.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03502-2
Received by editor(s): July 18, 1994
Additional Notes: The first author’s research was partially supported by RFFR Grant 93-011-1541 and ISF Grant MID 000. This paper was prepared while the second author was visiting the Institute for Advanced Study, whose hospitality is gratefully acknowledged
Communicated by: Ronald Solomon
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society