| Title:
|
Loewy coincident algebra and $QF$-3 associated graded algebra (English) |
| Author:
|
Tachikawa, Hiroyuki |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
59 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
583-589 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. \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. (English) |
| Keyword:
|
associated graded algebra |
| Keyword:
|
$QF$ algebra |
| Keyword:
|
$QF$-3 algebra |
| Keyword:
|
upper Loewy series |
| Keyword:
|
lower Loewy series |
| MSC:
|
13A30 |
| MSC:
|
16D50 |
| MSC:
|
16L60 |
| MSC:
|
16P70 |
| idZBL:
|
Zbl 1224.13007 |
| idMR:
|
MR2545641 |
| . |
| Date available:
|
2010-07-20T15:27:04Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140501 |
| . |
| Reference:
|
[1] Auslander, M.: Representation dimension of Artin algebras.Queen Mary College Lecture Notes (1971). Zbl 0331.16026 |
| Reference:
|
[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. Zbl 0080.25702, MR 0096700 |
| Reference:
|
[3] Nakayama, T.: On Frobeniusean algebras.II, Ann. Math. 42 (1941), 1-21. Zbl 0026.05801, MR 0004237, 10.2307/1968984 |
| Reference:
|
[4] Tachikawa, H.: Quasi-Frobenius rings and generalizations.LNM 351 (1973). Zbl 0271.16004 |
| Reference:
|
[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 2264126 |
| Reference:
|
[6] Thrall, R. M.: Some generalizations of quasi-Frobenius algebras.Trans. Amer. Math. Soc. 64 (1948), 173-183. Zbl 0041.01001, MR 0026048 |
| . |
