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)

 
 

 

Balanced and $ QF-1$ algebras


Authors: V. P. Camillo and K. R. Fuller
Journal: Proc. Amer. Math. Soc. 34 (1972), 373-378
MSC: Primary 16A36
DOI: https://doi.org/10.1090/S0002-9939-1972-0306256-0
MathSciNet review: 0306256
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A ring R is QF-1 if every faithful module has the double centralizer property. It is proved that a local finite dimensional algebra is QF-1 if and only if it is QF. From this it follows that an arbitrary finite dimensional algebra has the property that every homomorphic image is QF-1 if and only if every homomorphic image is QF.


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

  • [1] V. P. Camillo, Balanced rings and a problem of Thrall, Trans. Amer. Math. Soc. 149 (1970), 143-153. MR 41 #5417. MR 0260794 (41:5417)
  • [2] C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Appl. Math., vol. XI, Interscience, New York, 1962. MR 26 #2519. MR 0144979 (26:2519)
  • [3] S. E. Dickson and K. R. Fuller, Commutative QF-1 artinian rings are QF, Proc. Amer. Math. Soc. 24 (1970), 667-670. MR 40 #5646. MR 0252426 (40:5646)
  • [4] C. C. Faith, A general Wedderburn theorem, Bull. Amer. Math. Soc. 73 (1967), 65-67. MR 34 #2623. MR 0202763 (34:2623)
  • [5] D. R. Floyd, On QF-1 algebras, Pacific J. Math. 27 (1968), 81-94. MR 38 #3300. MR 0234988 (38:3300)
  • [6] K. R. Fuller, Generalized uniserial rings and their Kupisch series, Math. Z. 106 (1968), 248-260. MR 38 #1118. MR 0232795 (38:1118)
  • [7] -, Primary rings and double centralizers, Pacific J. Math. 34 (1970), 379-383. MR 42 #6040. MR 0271157 (42:6040)
  • [8] J. P. Jans, On the double centralizer condition, Math. Ann. 188 (1970), 85-89. MR 42 #7698. MR 0272817 (42:7698)
  • [9] K. Morita, Duality for modules and its application to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Sect. A 6 (1958), 83-142. MR 20 #3183. MR 0096700 (20:3183)
  • [10] -, On algebras for which every faithful representation is its own second commutator, Math. Z. 69 (1958), 429-434. MR 23 #A 3788. MR 0126492 (23:A3788)
  • [11] -, On S-rings in the sense of F. Kasch, Nagoya Math. J. 27 (1966), 687-695. MR 33 #7379. MR 0199230 (33:7379)
  • [12] K. Morita and H. Tachikawa, QF-3 rings (unpublished).
  • [13] T. Nakayama, On Frobeniusean algebras. II, Ann. of Math. (2) 42 (1941), 1-21. MR 2, 344. MR 0004237 (2:344b)
  • [14] -, Note on uni-serial and generalized uni-serial rings, Proc. Imp. Acad. Tokyo 16 (1940), 285-289. MR 2, 245. MR 0003618 (2:245c)
  • [15] C. J. Nesbitt and R. M. Thrall, Some ring theorems with applications to modular representations, Ann. of Math. (2) 47 (1946), 551-567. MR 8, 64. MR 0016760 (8:64a)
  • [16] R. M. Thrall, Some generalizations of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 64 (1948), 173-183. MR 10, 98. MR 0026048 (10:98c)
  • [17] T. Suzuki, A note on QF-1 algebras, Tôhoku J. Math. 22 (1970), 225-230. MR 0269697 (42:4592)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16A36

Retrieve articles in all journals with MSC: 16A36


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0306256-0
Keywords: Quasi-Frobenius, QF-1, finite dimensional algebra, double centralizer
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society