Loewy coincident algebra and QF-3 associated graded algebra
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.
Keywordsassociated graded algebra QF algebra QF-3 algebra upper Loewy series lower Loewy series
MSC 200013A30 16D50 16L60 16P70
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- M. Auslander: Representation dimension of Artin algebras. Queen Mary College Lecture Notes, 1971.Google Scholar
- 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
- H. Tachikawa: Quasi-Frobenius rings and generalizations. LNM 351, 1973.Google Scholar
- 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.