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Proceedings of the American Mathematical Society

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

 
 

 

Finite dimensional group rings


Author: Ralph W. Wilkerson
Journal: Proc. Amer. Math. Soc. 41 (1973), 10-16
MSC: Primary 16A26; Secondary 20C05
DOI: https://doi.org/10.1090/S0002-9939-1973-0318212-8
MathSciNet review: 0318212
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Abstract: A ring is right finite dimensional if it contains no infinite direct sum of right ideals. We prove that if a group $ G$ is finite, free abelian, or finitely generated abelian, then a ring $ R$ is right finite dimensional if and only if the group ring RG is right finite dimensional. A ring $ R$ is a self-injective cogenerator ring if $ {R_R}$ is injective and $ {R_R}$ is a cogenerator in the category of unital right $ R$-modules; this means that each right unital $ R$-module can be embedded in a direct product of copies of $ R$. Let $ G$ be a finite group where the order of $ G$ is a unit in $ R$. Then the group ring RG is a selfinjective cogenerator ring if and only if $ R$ is a self-injective cogenerator ring. Additional applications are given.


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

  • [1] W. D. Burgess, Rings of quotients of group rings, Canad. J. Math. 21 (1969), 865-875. MR 39 #5714. MR 0244399 (39:5714)
  • [2] I. G. Connell, On the group rings, Canad. J. Math. 15 (1963), 650-685. MR 27 #3666. MR 0153705 (27:3666)
  • [3] A. W. Goldie, Semiprime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201-220. MR 22 #2627. MR 0111766 (22:2627)
  • [4] J. Lambek, Lectures on rings and modules, Blaisdell, Waltham, Mass., 1966. MR 34 #5857. MR 0206032 (34:5857)
  • [5] D. S. Passman, Infinite group rings, Dekker, New York, 1971. MR 0314951 (47:3500)
  • [6] G. Renault, Sur les anneaux de groupes, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A84-A87. MR 44 #5387. MR 0288189 (44:5387)
  • [7] P. Ribenboim, Rings and modules, Interscience Tracts in Pure and Appl. Math., no. 24, Interscience, New York, 1969. MR 39 #4204. MR 0242877 (39:4204)
  • [8] R. C. Shock, Injectivity, annihilators, and orders, J. Algebra 19 (1971), 96-103. MR 43 #4861. MR 0279135 (43:4861)
  • [9] -, Orders in self-injective cogenerator rings, Proc. Amer. Math. Soc. 35 (1972), 393-398. MR 0302683 (46:1827)
  • [10] -, Polynomial rings over finite dimensional rings, Pacific J. Math. 42 (1972), 251-258. MR 0318201 (47:6748)
  • [11] L. W. Small, Orders in Artinian rings. II, J. Algebra 9 (1968), 266-273. MR 37 #6315. MR 0230755 (37:6315)
  • [12] P. F. Smith, Quotient rings of group rings, J. London Math. Soc. 3 (1971), 645-660. MR 0314886 (47:3435)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0318212-8
Keywords: Group ring, injective, order, cogenerator, rationally closed, dense right ideal, complete ring of quotients, right finite dimensional
Article copyright: © Copyright 1973 American Mathematical Society

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