Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the semigroups of fully invariant ideals of the free group algebra and the free associative algebra
HTML articles powered by AMS MathViewer

by Samuel M. Vovsi
Proc. Amer. Math. Soc. 119 (1993), 1029-1037
DOI: https://doi.org/10.1090/S0002-9939-1993-1154250-6
PDF | Request permission

Abstract:

Let $R$ be an integral domain, $K$ its field of fractions, $F$ a free group. Let $I$ and $J$ be fully invariant (=verbal) ideals of the group algebra $KF$. We prove that over certain domains the equality $IJ \cap RF = (I \cap RF) \times (J \cap RF)$ need not be true. A similar result is valid for fully invariant ideals of the free associative algebra. This implies that the product of pure varieties of group representations over an integral domain need not be pure, that there exist pure nonprojective varieties of group representations and of associative algebras, and also answers some other questions raised in the literature.
References
    G. M. Bergman, Personal communication, January 1983.
  • George M. Bergman and Jacques Lewin, The semigroup of ideals of a fir is (usually) free, J. London Math. Soc. (2) 11 (1975), no. 1, 21–31. MR 379554, DOI 10.1112/jlms/s2-11.1.21
  • Wilhelm Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann. 111 (1935), no. 1, 259–280 (German). MR 1512992, DOI 10.1007/BF01472217
  • B. I. Plotkin, Varieties of group representations, Uspehi Mat. Nauk 32 (1977), no. 5 (197), 3–68, 239 (Russian). MR 0470082
  • B. I. Plotkin and A. S. Grinberg, Semigroups of varieties that are connected with group representations, Sibirsk. Mat. Ž. 13 (1972), 841–858 (Russian). MR 0306326
  • B. I. Plotkin and S. M. Vovsi, Mnogoobraziya predstavleniĭ grupp, “Zinatne”, Riga, 1983 (Russian). Obshchaya teoriya, svyazi i prilozheniya. [General theory, connections and applications]. MR 739330
  • S. M. Vovsi, Varieties and triangular products of projective representations of groups, Sibirsk. Mat. Zh. 21 (1980), no. 3, 56–62, 236 (Russian). MR 574962
    • —, Triangular products of group representations and their applications, Progr. in Math., vol. 17, Birkhäuser, Boston, MA, 1981.
  • S. M. Vovsi, Triangular products and identities of some associative algebras, Algebra i Logika 26 (1987), no. 1, 27–35, 125 (Russian). MR 950260
  • S. M. Vovsi, Homogeneous varieties of representations of groups, Sibirsk. Mat. Zh. 29 (1988), no. 3, 35–47, 219 (Russian); English transl., Siberian Math. J. 29 (1988), no. 3, 360–371 (1989). MR 953019, DOI 10.1007/BF00969643
  • Samuel M. Vovsi, Topics in varieties of group representations, London Mathematical Society Lecture Note Series, vol. 163, Cambridge University Press, Cambridge, 1991. MR 1149131, DOI 10.1017/CBO9780511525964
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16S34, 20C07
  • Retrieve articles in all journals with MSC: 16S34, 20C07
Bibliographic Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 119 (1993), 1029-1037
  • MSC: Primary 16S34; Secondary 20C07
  • DOI: https://doi.org/10.1090/S0002-9939-1993-1154250-6
  • MathSciNet review: 1154250