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On perfect simple-injective rings


Authors: W. K. Nicholson and M. F. Yousif
Journal: Proc. Amer. Math. Soc. 125 (1997), 979-985
MSC (1991): Primary 16D50, 16L30
DOI: https://doi.org/10.1090/S0002-9939-97-03678-2
MathSciNet review: 1363179
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Abstract: Harada calls a ring $R$ right simple-injective if every $R$-homomorphism with simple image from a right ideal of $R$ to $R$ is given by left multiplication by an element of $R$. In this paper we show that every left perfect, left and right simple-injective ring is quasi-Frobenius, extending a well known result of Osofsky on self-injective rings. It is also shown that if $R$ is left perfect and right simple-injective, then $R$ is quasi-Frobenius if and only if the second socle of $R$ is countably generated as a left $R$-module, extending many recent results on self-injective rings. Examples are given to show that our results are non-trivial extensions of those on self-injective rings.


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

  • 1. P. Ara, and J. K. Park, On continuous semiprimary rings, Comm. Alg. 19 (1991), 1945-1957. MR 92g:16006
  • 2. E. P. Armendariz and J. K. Park, , Arch. Math. 58 (1992), 24-33. MR 92m:16002
  • 3. Y. Baba and K. Oshiro, On a Theorem of Fuller, J. Algebra, 154 (1993), 86-94. MR 94b:16025
  • 4. J. A. Beachy, On quasi-artinian rings, J. London Math. Soc. (2) 3 (1971), 449-452. MR 44:244
  • 5. J.-E. Björk, Rings satisfying certain chain conditions, J. Reine Angew. Math. 245 (1970), 63-73. MR 43:3295
  • 6. N. Bourbaki, ``Algebra, Part I'', Hermann/Addison Wesley, 1974, p. 400. MR 50:6689
  • 7. V. Camillo, Commutative rings whose principal ideals are annihilators, Portugaliae Math. 46 (1) (1989), 33-37. MR 90e:13003
  • 8. J. Clark and D. V. Huynh, A note on perfect selfinjective rings, Quart. J. Math. Oxford (2) 45 (1994), 13-17. MR 95a:16005
  • 9. C. Faith, When self-injective rings are QF: A report on a problem, Centre Recerca Matemàtica Institut d'Estudis Catalans (Spain), 1990.
  • 10. K. R. Fuller, On indecomposable injectives over artinian rings, Pacific J. Math. 29 (1969), 115-135. MR 40:186
  • 11. C. R. Hajarnavis and N. C. Norton, On dual rings and their modules, J. Algebra 93 (1985), 253-266. MR 86i:16016
  • 12. M. Harada, On almost relative injective of finite length, preprint.
  • 13. L. S. Levy, Commutative rings whose homomorphic images are self-injective, Pacific J. Math. 18 (1966), 149-153. MR 33:2663
  • 14. W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra 174 (1995), 77-93. MR 96i:16005
  • 15. B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373-389. MR 34:4305; MR 36:6443

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Additional Information

W. K. Nicholson
Affiliation: Department of Mathematics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
Email: wknichol@acs.ucalgary.ca

M. F. Yousif
Affiliation: Department of Mathematics, Ohio State University, Lima, Ohio 45804
Email: yousif.1@osu.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03678-2
Keywords: Perfect ring, self-injective ring, quasi-Frobenius ring
Received by editor(s): April 24, 1995
Received by editor(s) in revised form: October 11, 1995
Additional Notes: The research of both authors was supported by NSERC Grant 8075 and by the Ohio State University.
Dedicated: Dedicated to Professor K. Varadarajan on the occasion of his sixtieth birthday
Communicated by: Ken Goodearl
Article copyright: © Copyright 1997 American Mathematical Society

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