Some Noetherian rings

McConnell, J. C.; Robson, J. C.

doi:10.1090/gsm/030/02

Cited by 111 publications

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“…Since D is a noetherian domain, D satisfies the left and the right Ore properties (see, e.g., [30]), namely, for all d 1 , d 2 ∈ D \ {0}, there exist e 1 , e 2 ∈ D \ {0} such that e 1 d 1 = e 2 d 2 (resp., d 1 e 1 = d 2 e 2 ). This result yields the existence of the division ring of fractions…”

Section: Similar Definitions Hold For Right D-modulesmentioning

confidence: 99%

“…, noetherian and satisfies the following property: Hence, every two-sided ideal of D is trivial, i.e., D is a simple ring (see, e.g., [30,38]). …”

Section: Very Simple Domainsmentioning

confidence: 99%

“…A well-known result states that a left (resp., right) noetherian domain satisfies the left (resp., right) Ore condition (see, e.g., [24,30]). Hence, if D is a very simple domain and Section 2).…”

Section: Very Simple Domainsmentioning

confidence: 99%

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A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Quadrat

Robertz

2014

Acta Appl Math

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To cite this version:Alban Quadrat, Daniel Robertz. A constructive study of the module structure of rings of partial differential operators. [ Abstract:The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two to two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r + 1 elements but no fewer. These results are implemented in the Stafford package for D = A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series. Key-words:Weyl algebras, Stafford's theorems, linear systems of partial differential equations, D-modules, mathematical systems theory, constructive algebra, symbolic computation. A constructive study of the module structure of rings of partial differential operators 4

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Section: Similar Definitions Hold For Right D-modulesmentioning

confidence: 99%

“…, noetherian and satisfies the following property: Hence, every two-sided ideal of D is trivial, i.e., D is a simple ring (see, e.g., [30,38]). …”

Section: Very Simple Domainsmentioning

confidence: 99%

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A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Quadrat

Robertz

2014

Acta Appl Math

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“…Morita equivalence and Morita context. Let us recall some facts we need on Morita equivalence, for proofs the reader is refereed to [8]: if algebras A and B are Morita equivalent then GK (A) = GK (B); A is a simple algebra iff B is a simple algebra; A is a finitely generated algebra iff B is a finitely generated algebra. …”

Section: Gk (M) ≥ Gk (A) Fdim (A) + Max{fdim (A) 1}mentioning

confidence: 99%

“…The algebra B is Morita equivalent to A, [8] (the Morita context A Ae eA eAe provides a Morita equivalence since eAAe = eAe and AeeA = A, since A is simple and e = 0). Now, applying Corollary 2.1 we have the result.…”

Section: Gk (M) ≥ H a ≥ Gk (A) Fdim (A) + Max{fdim (A) 1}mentioning

confidence: 99%

Morita Invariance of the Filter Dimension and of the Inequality of Bernstein

Bavula

Hinchcliffe

2008

Algebr Represent Theor

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It is proved that the filter dimenion is Morita invariant. A direct consequence of this fact is the Morita invariance of the inequality of Bernstein: if an algebra A is Morita equivalent to the ring D(X) of differential operators on a smooth irreducible affine algebraic variety X of dimension n ≥ 1 over a field of characteristic zero then the Gelfand-Kirillov dimension GK (M ) ≥ n = GK (A) 2 for all nonzero finitely generated A-modules M . In fact, a more strong result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. As a direct consequence of this fact an affirmative answer is given to the question/conjecture posed by Ken Brown several years ago of whether an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras H c for integral c: GK (M ) ≥ n = GK (Hc) 2 for all nonzero finitely generated H c -modules M .

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Noetherian Semigroup Algebras

Jespers

Okniński

2007

Algebra and Applications

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Algebra and Applications aims to publish well written and carefully refereed monographs with up-to-date information about progress in all fields of algebra, its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, as well as applications in related domains, such as number theory, homotopy and (co)homology theory, physics and discrete mathematics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, C*algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications. In addition, Algebra and Applications will also publish monographs dedicated to computational aspects of these topics as well as algebraic and geometric methods in computer science.

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Some Noetherian rings

Cited by 111 publications

References 0 publications

A Constructive Study of the Module Structure of Rings of Partial Differential Operators

A Constructive Study of the Module Structure of Rings of Partial Differential Operators

Morita Invariance of the Filter Dimension and of the Inequality of Bernstein

Noetherian Semigroup Algebras

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