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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  1. Auslander, M., Representation dimension of Artin algebras, Queen Mary College Lecture Notes (1971). (1971) Zbl0331.16026
  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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