A generalization of heredity ideals

Hoshino, Mitsuo; Yukimoto, Yoshito

doi:10.21099/tkbjm/1496161463

Cited by 3 publications

(4 citation statements)

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“…The following proposition generalizes the results stated in [21, (1.4)] and [4, (1.3)]. We refer to [5,17,23] for more similar matrix reductions. For convenience, we define cd(0) = 1.…”

Section: )] Which Is Called the Cartan Determinant Of A And Denoted supporting

confidence: 79%

Some homological conjectures for quasi-stratified algebras

Liu¹,

Paquette²

2006

Journal of Algebra

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“…The following proposition generalizes the results stated in [21, (1.4)] and [4, (1.3)]. We refer to [5,17,23] for more similar matrix reductions. For convenience, we define cd(0) = 1.…”

Section: )] Which Is Called the Cartan Determinant Of A And Denoted supporting

confidence: 79%

Some homological conjectures for quasi-stratified algebras

Liu¹,

Paquette²

2006

Journal of Algebra

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“…In the case when 11 is projective, the assumption that Torr (Aft, 11) = 0 forj __> 1 holds obviously, but the fact that det C (A/I) = det C (A) is trivial because AI belongs to add (11) (see the first paragraph of 2.2). In the case when I~ is not projective, Hoshino-Yukimoto [9] …”

Section: Proposition Det C T (A/i) = Det Ccrmentioning

confidence: 99%

A reduction formula for the Cartan determinant problem for algebras

Yamagata

1993

Arch. Math

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In 1958 Eilenberg [5] showed that the Cartan matrix C (A) of a left artinian ring A of finite (left) global dimension must satisfy det C(A) = + 1. But it is not yet known whether there are any left artinian rings of finite global dimension whose Cartan determinants are -1. The Cartan determinant problem is the question whether for a left artinian ring of finite global dimension the determinant of the Cartan matrix is necessarily 1. Concerning this problem it is known that the determinant of the Cartan matrix is I for any of the following left artinian rings of finite global dimension: positively graded algebras over a field [10], left artinian rings whose radical cube is zero or, more generally, Cartan filtered rings [7], left artinian rings whose global dimension is smaller than 3 Moreover, we shall show how this relation implies the results mentioned above. Throughout this paper, alI rings are left artinian basic rings, and modules are finitely generated left modules.

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“…We suppose for the moment that R is left artinian and gl dim R < oo. The first method was initiated by Zacharia in [12] and was used in [4], [9] and [1]. Here the idea is to find e 2 = e e R so that det C(R) = det C(eRe) and gl dim eRe < oo.…”

Section: Introductionmentioning

confidence: 99%

“…The easiest case is where / = ReR, for some idempotent e, is an heredity ideal (see, e.g., [3]). Generalizations of this method are found in [9] and [11].…”

Section: Introductionmentioning

confidence: 99%

The Cartan determinant and generalizations of quasihereditary rings

Burgess

Fuller

1998

Proceedings of the Edinburgh Mathematical Society

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The Cartan determinant conjecture for left artinian rings was verified for quasihereditary rings showing detC(R) = detC(U//), where / is a projective ideal generated by a primitive idempotent. This article identifies classes of rings generalizing the quasihereditary ones, first by relaxing the "projective" condition on heredity ideals. These rings, called left fc-hereditary are all of finite global dimension. Next a class of rings is defined which includes left serial rings of finite global dimension, quasihereditary and left 1-hereditary rings, but also rings of infinite global dimension. For such rings, the Cartan determinant conjecture is true, as is its converse. This is shown by matrix reduction. Examples compare and contrast these rings with other known families and a recipe is given for constructing them.

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A generalization of heredity ideals

Cited by 3 publications

References 0 publications

Some homological conjectures for quasi-stratified algebras

Some homological conjectures for quasi-stratified algebras

A reduction formula for the Cartan determinant problem for algebras

The Cartan determinant and generalizations of quasihereditary rings

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