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Loewy coincident algebra and $QF$-3 associated graded algebra
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 583-589
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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. \endgraf $QF$-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.
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. \endgraf $QF$-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.
Classification : 13A30, 16D50, 16L60, 16P70
Keywords: associated graded algebra; $QF$ algebra; $QF$-3 algebra; upper Loewy series; lower Loewy series 
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     author = {Tachikawa, Hiroyuki},
     title = {Loewy coincident algebra and $QF$-3 associated graded algebra},
     journal = {Czechoslovak Mathematical Journal},
     pages = {583--589},
     year = {2009},
     volume = {59},
     number = {3},
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     zbl = {1224.13007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a1/}
}
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Tachikawa, Hiroyuki. Loewy coincident algebra and $QF$-3 associated graded algebra. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 583-589. http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a1/

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