Loewy coincident algebra and \(QF\)-\(3\) associated graded algebra. (English) Zbl 1224.13007

Summary: 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.


13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
16D50 Injective modules, self-injective associative rings
16L60 Quasi-Frobenius rings
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
Full Text: DOI EuDML


[1] M. Auslander: Representation dimension of Artin algebras. Queen Mary College Lecture Notes, 1971.
[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. · Zbl 0080.25702
[3] T. Nakayama: On Frobeniusean algebras. II, Ann. Math. 42 (1941), 1–21. · Zbl 0026.05801 · doi:10.2307/1968984
[4] H. Tachikawa: Quasi-Frobenius rings and generalizations. LNM 351, 1973. · Zbl 0271.16004
[5] H. Tachikawa: QF rings and QF associated graded rings. Proc. 38th Symposium on Ring Theory and Representation Theory, Japan, pp. 45–51. http://fuji.cec.yamanash.ac.jp/ring/oldmeeting/2005/reprint2005/abst-3-2.pdf .
[6] R. M. Thrall: Some generalizations of quasi-Frobenius algebras. Trans. Amer. Math. Soc. 64 (1948), 173–183. · Zbl 0041.01001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.