A Note on a Class of Simple Noetherian Domains

Shamsuddin, Ahmad

doi:10.1112/jlms/s2-15.2.213

Cited by 9 publications

(2 citation statements)

References 2 publications

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“…Given a simple Noetherian ring that is "built up" from simple rings of Krull dimension one, it ought to be possible to repeat the arguments of this paper. For example this is possible for Shamsuddin's example of a simple Noetherian ring of infinite Krull dimension [8]. So any right ideal of this ring can be generated by just two elements.…”

Section: Comments and Questionsmentioning

confidence: 99%

Module Structure of Weyl Algebras

Stafford

1978

Journal of the London Mathematical Society

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Define the «-th Weyl algebra, A n (R), over a ring R to be the associative tf-algebra, with identity, generated by the In indeterminates x u ...,x n , 0 x , ...,0 n subject to the relations:XiXj-XjXi = 0 = Ofij-OjOi and xfij-OjXi = d u .This ring has been much studied in the case when R is a field k of characteristic zero. For the special case of A t (k), it has been shown that, in particular, any right ideal can be generated by two elements [1] and that any projective right module is either free or isomorphic to a right ideal [13]. The intention of this paper is to show that the same results are true for ^n(A:)-modules. This extends the results of [9], where it was shown that, for example, any right ideal of A n can be generated by n + ] elements, and those of [10], which reduced this bound from n+1 to five. Specifically, in Section 3, we prove:(1) Any right ideal of A n = A n (k) can be generated by two elements. Moreover if a right ideal I = aA n + bA n + cA n , and d ^ 0eA n , then there exist / and geA n such that / = (a + cfd)A n + (b + cgd)A n .

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Section: Comments and Questionsmentioning

confidence: 99%

Module Structure of Weyl Algebras

Stafford

1978

Journal of the London Mathematical Society

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“…As the first step of proving these conjectures, people try to see if these simple rings are left V-rings and then proceed from there. Simple rings are interesting objects of study with a vast literature about them; this is especially true of simple Noetherian rings (see [11,38,5,6,34,32,35,36,7,21,19,23,3,4], amongst others). Dealing with simple rings is not as easy a task as it may sound.…”

Section: Introductionmentioning

confidence: 99%

Some results on V-rings and weakly V-rings

Dinh¹,

Holston²,

Huynh³

2013

Journal of Pure and Applied Algebra

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a b s t r a c tA ring R is called a right weakly V-ring (briefly, a right WV-ring) if every simple right R-module is X -injective, where X is any cyclic right R-module with X R R R . In this note, we study the structure of right WV-rings R and show that, if R is not a right V-ring, then R has exactly three distinct ideals, 0 ⊂ J ⊂ R, where J is a nilpotent minimal right ideal of R such that R/J is a simple right V-domain. In this case, if we assume additionally that R J is finitely generated, then R is left Artinian and right uniserial with composition length 2. We also show that a strictly right WV-ring with Jacobson radical J is a Frobenius local ring if and only if the injective hull of J R is uniserial. Some other results are obtained in the connection with the Noetherian property of right WV-rings and related rings.

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