Geodesic
Loewy coincident algebra and $QF$-3 associated graded algebra
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 583-589 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 
@article{CMJ_2009_59_3_a1,
     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},
     mrnumber = {2545641},
     zbl = {1224.13007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a1/}
}
TY  - JOUR
AU  - Tachikawa, Hiroyuki
TI  - Loewy coincident algebra and $QF$-3 associated graded algebra
JO  - Czechoslovak Mathematical Journal
PY  - 2009
SP  - 583
EP  - 589
VL  - 59
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a1/
LA  - en
ID  - CMJ_2009_59_3_a1
ER  - 
%0 Journal Article
%A Tachikawa, Hiroyuki
%T Loewy coincident algebra and $QF$-3 associated graded algebra
%J Czechoslovak Mathematical Journal
%D 2009
%P 583-589
%V 59
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2009_59_3_a1/
%G en
%F CMJ_2009_59_3_a1
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/

[1] Auslander, M.: Representation dimension of Artin algebras. Queen Mary College Lecture Notes (1971). | Zbl

[2] Morita, K.: 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. | MR | Zbl

[3] Nakayama, T.: On Frobeniusean algebras. II, Ann. Math. 42 (1941), 1-21. | DOI | MR | Zbl

[4] Tachikawa, H.: Quasi-Frobenius rings and generalizations. LNM 351 (1973). | Zbl

[5] Tachikawa, H.: QF rings and QF associated graded rings. Proc. 38th Symposium on Ring Theory and Representation Theory, Japan 45-51.\hfil http://fuji.cec.yamanash.ac.jp/ring/oldmeeting/2005/reprint2005/abst-3-2.pdf | MR

[6] Thrall, R. M.: Some generalizations of quasi-Frobenius algebras. Trans. Amer. Math. Soc. 64 (1948), 173-183. | MR | Zbl