A general notion of noncommutative Krull rings

Jespers, Eric; Wauters, P.

doi:10.1016/0021-8693(88)90098-1

Cited by 12 publications

(9 citation statements)

References 15 publications

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“…We recall that that is, the direct limit of the system PROOF. This result has been proved in [9], using Proposition 1.1. But here is a short direct proof.…”

Section: Cancellative Semigroup Graded Ringsmentioning

confidence: 78%

Chain conditions and semigroup graded rings

Jespers¹

1988

J Aust Math Soc A

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“…We recall that that is, the direct limit of the system PROOF. This result has been proved in [9], using Proposition 1.1. But here is a short direct proof.…”

Section: Cancellative Semigroup Graded Ringsmentioning

confidence: 78%

Chain conditions and semigroup graded rings

Jespers¹

1988

J Aust Math Soc A

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“…Because G satisfies the ascending chain condition on cyclic subgroups, by Matsuda's result, we know that R e = K[G ] is a Krull domain. Hence, by the corollary mentioned before (see [11]) and the properties stated in the introduction, R = K[S] is a Krull order if and only if G ⊆ S and S/G is a Krull monoid, or equivalently, S is a Krull monoid.…”

Section: Corollary 1 If G/g Is Torsion-free and Both G And G/g Satismentioning

confidence: 97%

“…For definitions on maximal orders and Krull orders, we refer to [7,11]. So, a prime (left and right) Goldie ring R with classical ring of quotients Q is said to be a Krull order in Q if it is a maximal order that satisfies the ascending chain condition on integral divisorial ideals.…”

mentioning

confidence: 99%

“…[11,Corollary 4.7] implies that, for the strongly G/G -graded ring R, if G/G satisfies the ascending chain condition on cyclic subgroups and R e = K[G ] is a Krull order, then the subring K[S] = R [S/G ] is a Krull order if and only if S/G is a Krull monoid. The proof of this result is based on a result of Chouinard II [3], stating that a commutative monoid ring K[A] is a Krull domain if and only if A is a Krull monoid such that its group of fractions satisfies the ascending chain condition on cyclic subgroups.…”

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confidence: 99%

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Krull orders in nilpotent groups: corrigendum and addendum

Jespers

Okniński

2016

Arch. Math.

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Noncommutative Krull domains that are determined by submonoids of torsion-free nilpotent groups are investigated. A complete description is given in case the group G is nilpotent of class two and its abelianisation is torsion-free and satisfies the ascending chain condition on cyclic subgroups. The result corrects and extends an earlier result by the authors to the case that G is not necessarily finitely generated and yields a class of non-Noetherian algebras that have a nice arithmetical structure.In [7] the authors investigate the arithmetical properties, such as being a Krull order, of a class of algebras that are non-Noetherian, which is motivated by deformations of some quadratic algebras showing up in the classification of four dimensional noncommutative projective surfaces, see [13,14]. The algebras under consideration turn out to be semigroup algebras K[S] over a field K, where S is a submonoid of a torsion-free nilpotent group of class 2.For definitions on maximal orders and Krull orders, we refer to [7,11]. So, a prime (left and right) Goldie ring R with classical ring of quotients Q is said to be a Krull order in Q if it is a maximal order that satisfies the ascending chain condition on integral divisorial ideals.The main result in [7], namely Theorem 3.4, is a characterization of when such an algebra K[S] is a Krull order (note that it is a domain as S is a submonoid of a torsion-free nilpotent group). It is shown that if

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“…Since R is prime Goldie it follows that Re is semiprime Goldie (see for example [8]). It is then well known (see for example [9]) that Qg = Qd(Rg) = CJX Re = ReCJx .…”

Section: Qgcc~xrnrmentioning

confidence: 99%