d-Simple rings and simple -modules

Coutinho, S. C.

doi:10.1017/s0305004198002965

Cited by 19 publications

(14 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, r + s), the polynomial P (a,b j ) (X) (respectively P (−a,b j ) (X)) can be computed. Then, of course, one can effectively determine whether the properties (ii)(0)- (2) are satisfied.…”

Section: Theorem B (The Algorithm In the ''Local'' Case) Let Rmentioning

confidence: 99%

Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization

Lequain

2011

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

a b s t r a c tLet K be a field of characteristic zero, n ≥ 1 an integer and A n+1 = ⟩ the (n+1)th Weyl algebra over K . Let S ∈ A n+1 be an order-1 differential operator of the typeWe construct an algorithm that allows one to recognize whether S generates a maximal left ideal of A n+1 , hence also whether A n+1 /A n+1 S is an irreducible non-holonomic A n+1 -module. The algorithm, which is a powerful instrument for producing concrete examples of cyclic maximal left ideals of A n , is easy to implement and quite useful; we use it to solve several open questions.The algorithm also allows one to recognize whether certain families of algebraic differential equations have a solution in K [X, Y 1 , . . . , Y n ] and, when they have one, to compute it.

show abstract

“…, r + s), the polynomial P (a,b j ) (X) (respectively P (−a,b j ) (X)) can be computed. Then, of course, one can effectively determine whether the properties (ii)(0)- (2) are satisfied.…”

Section: Theorem B (The Algorithm In the ''Local'' Case) Let Rmentioning

confidence: 99%

Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization

Lequain

2011

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

show abstract

“…, x n ] are not easy to find. A family of linear simple derivations was discovered by Coutinho in [3] (generalizing an example of Stafford) and is based on a result of Shamsuddin (see [10]). Families of simple quadratic derivations of K[x 1 , x 2 ] were found by Maciejewski, Moulin-Ollagnier and Nowicki in [9].…”

Section: Proof It Is Enough To Show That (Ii) ⇒ (I) Suppose Thatmentioning

confidence: 99%

“…Stafford's examples were generalized by Coutinho in [3]. Another objective also treated in section 3 is to generalize to K[x 1 , .…”

Section: Introduction Letmentioning

confidence: 99%

D-Simple Rings and Principal Maximal Ideals of the Weyl Algebra

2005

View full text Add to dashboard Cite

show abstract

“…More recently, the d-simplicity of the ring of polynomials has also been used to produce new examples of nonholonomic irreducible modules over the Weyl algebra; see [4,7].…”

Section: Introductionmentioning

confidence: 99%