Loewy coincident algebra and Q F -3 associated graded algebra

Hiroyuki Tachikawa

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 3, page 583-589
  • ISSN: 0011-4642

Abstract

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We prove that an associated graded algebra R G of a finite dimensional algebra R is Q F (= selfinjective) if and only if R is Q F 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 R e and e R coincide. Q F -3 algebras are an important generalization of Q F 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 Q F -3 if and only if R is Q F -3.

How to cite

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Tachikawa, Hiroyuki. "Loewy coincident algebra and $QF$-3 associated graded algebra." Czechoslovak Mathematical Journal 59.3 (2009): 583-589. <http://eudml.org/doc/37943>.

@article{Tachikawa2009,
abstract = {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.},
author = {Tachikawa, Hiroyuki},
journal = {Czechoslovak Mathematical Journal},
keywords = {associated graded algebra; $QF$ algebra; $QF$-3 algebra; upper Loewy series; lower Loewy series; associated graded algebra; algebra; -3 algebra; upper Loewy series; lower Loewy series},
language = {eng},
number = {3},
pages = {583-589},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Loewy coincident algebra and $QF$-3 associated graded algebra},
url = {http://eudml.org/doc/37943},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Tachikawa, Hiroyuki
TI - Loewy coincident algebra and $QF$-3 associated graded algebra
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 3
SP - 583
EP - 589
AB - 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.
LA - eng
KW - associated graded algebra; $QF$ algebra; $QF$-3 algebra; upper Loewy series; lower Loewy series; associated graded algebra; algebra; -3 algebra; upper Loewy series; lower Loewy series
UR - http://eudml.org/doc/37943
ER -

References

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  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. (1958) Zbl0080.25702MR0096700
  3. Nakayama, T., 10.2307/1968984, II, Ann. Math. 42 (1941), 1-21. (1941) Zbl0026.05801MR0004237DOI10.2307/1968984
  4. Tachikawa, H., Quasi-Frobenius rings and generalizations, LNM 351 (1973). (1973) Zbl0271.16004
  5. Tachikawa, H., QF rings and QF associated graded rings, Proc. 38th Symposium on Ring Theory and Representation Theory, Japan 45-51.http://fuji.cec.yamanash.ac.jp/ring/oldmeeting/2005/reprint2005/abst-3-2.pdf. MR2264126
  6. Thrall, R. M., Some generalizations of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 64 (1948), 173-183. (1948) Zbl0041.01001MR0026048

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