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

**[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**

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