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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.

MSC:

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)
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References:

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