Global dimension in serial rings

Gustafson, William H.

doi:10.1016/0021-8693(85)90069-9

Cited by 45 publications

(46 citation statements)

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“…Thus we may assume that the quiver of R is a cycle and, hence, that R has a Kupisch series, Re ,Re n . Methods and notation from Gustafson [8] Also, by [8, (6)], if i,j e X, any homomorphism g : Re, -* Re t is either 0 or is an isomorphism. Hence, in particular, if i e X, LL (Re,) < n.…”

Section: Proposition 23 If R Is a Left Serial Artinian Ring Which Imentioning

confidence: 99%

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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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Section: Proposition 23 If R Is a Left Serial Artinian Ring Which Imentioning

confidence: 99%

“…The series e n , e,,..., e n _ x is matrix reducing. This is the family of examples of [8], which exhibit maximal possible global dimension for a serial ring with n idempotents.…”

Section: Examplesmentioning

confidence: 99%

The Cartan determinant and generalizations of quasihereditary rings

Burgess

Fuller

1998

Proceedings of the Edinburgh Mathematical Society

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“…A theorem that describes properties of Artinian serial rings with finite global dimension was proved in [2]. The main result of the present paper is the construction of Artinian serial rings that satisfy the conditions of the main theorem in [2] (beginning with a tiled order of width 1) and tiled orders of width two with maximal global dimension (using the methods given in [3]).…”

Section: Introductionmentioning

confidence: 96%

Serial rings and tiled orders of width two

Bronitskaya

2009

Ukr Math J

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“…It follows that Rek/Jek has projective dimension 1. D One direction of the equivalence in the next lemma is due to Zacharia [14,Lemma 2]; the inequality appears in Gustaf son [7] with the added hypothesis that R be serial and gl dim R < oo. In establishing the bridge between Cartan matrix and global dimension, we will use it to successively reduce the number of primitive idempotents.…”

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confidence: 99%