Czechoslovak Mathematical Journal

, Volume 59, Issue 3, pp 583–589 | Cite as

Loewy coincident algebra and QF-3 associated graded algebra

  • Hiroyuki Tachikawa


We prove that an associated graded algebra R G of a finite dimensional algebra R is QF (= selfinjective) if and only if R is QF and Loewy coincident. Here R is said to be Loewy coincident if, for every primitive idempotent e, the upper Loewy series and the lower Loewy series of Re and eR coincide. QF-3 algebras are an important generalization of QF algebras; note that Auslander algebras form a special class of these algebras. We prove that for a Loewy coincident algebra R, the associated graded algebra R G is QF-3 if and only if R is QF-3.


associated graded algebra QF algebra QF-3 algebra upper Loewy series lower Loewy series 

MSC 2000

13A30 16D50 16L60 16P70 


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  1. [1]
    M. Auslander: Representation dimension of Artin algebras. Queen Mary College Lecture Notes, 1971.Google Scholar
  2. [2]
    K. Morita: Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A. No. 150 (1958), 1–60.Google Scholar
  3. [3]
    T. Nakayama: On Frobeniusean algebras. II, Ann. Math. 42 (1941), 1–21.CrossRefMathSciNetGoogle Scholar
  4. [4]
    H. Tachikawa: Quasi-Frobenius rings and generalizations. LNM 351, 1973.Google Scholar
  5. [5]
    H. Tachikawa: QF rings and QF associated graded rings. Proc. 38th Symposium on Ring Theory and Representation Theory, Japan, pp. 45–51.˜ring/oldmeeting/2005/reprint2005/abst-3-2.pdf.
  6. [6]
    R. M. Thrall: Some generalizations of quasi-Frobenius algebras. Trans. Amer. Math. Soc. 64 (1948), 173–183.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2009

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of TsukubaTsukuba, IbarakiJapan

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