Wild torsion modules over Weyl algebras and general torsion modules over HNPs

Klingler, Lee; Levy, Lawrence S.

doi:10.1016/s0021-8693(05)80003-1

Cited by 11 publications

(14 citation statements)

References 28 publications

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“…Since the main thrust of the present paper is that the theory of projective modules over HNP rings echoes the classical theory of commutative Dedekind domains more than previously suspected, it seems appropriate to mention some related results in this same vein, from [14]. As above, R denotes an arbitrary HNP ring.…”

Section: Contextmentioning

confidence: 81%

Hereditary Noetherian Prime Rings 2. Finitely Generated Projective Modules

Levy

Robson

1999

Journal of Algebra

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Let R be a hereditary Noetherian prime ring. We determine a full set of invariants for the isomorphism class of any finitely generated projective R-module of uniform dimension at least 2. In particular we prove that P ⊕ X ∼ = Q ⊕ X implies P ∼ = Q whenever P has uniform dimension at least 2. Among the applications of these results are necessary and sufficient conditions for the existence of a bound to the number of generators needed for right ideals of R.

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Section: Contextmentioning

confidence: 81%

Hereditary Noetherian Prime Rings 2. Finitely Generated Projective Modules

Levy

Robson

1999

Journal of Algebra

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“…The situation for the algebra B,, which is a differential polynomial ring, is similar. This is done using a construction of Klingler and Levy [20] which they used for proving wildness of the category of finite length modules over A~. Supporting this point of view we prove that the theory of modules over a wide class of generalized Weyl algebras interprets the word problem for groups.…”

mentioning

confidence: 72%

“…In [20] Klingler and Levy, starting with a special sextuple of simple modules over the first Weyl algebra At(k) construct a map from the category of modules over the free associative algebra k<X,Y} to the category of modules over A~ with very nice properties (e.g., this map preserves endomorphism rings). The proof of the following result involves this construction very heavily.…”

Section: Decidabilitymentioning

confidence: 99%

Some model theory over hereditary noetherian domains

Prest

Puninski

1999

Journal of Algebra

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Questions in the model theory of modules over hereditary noetherian domains are investigated with particular attention being paid to differential polynomial rings and to generalized Weyl algebras. We prove that there exists no isolated point in the Ziegler spectrum over a simple hereditary generalized Weyl algebra A of the sort considered by Bavula [Algebra iAnaliz 4(1) (1992), 75 97] over a field k with char(iv) = 0 (the first Weyl algebra Al(k) is one such) and the category of finite length modules over A does not have any almost split sequence. We show that the theory of all modules over a wide class of generalized Weyl algebras and related rings interprets the word problem for groups, and in the case that the field is countable there exists a superdecomposable pure-injective module over A. This class includes, for example, the universal enveloping algebra Usl2(k). ~, 1099Academic Press PRELIMINARIESA main theme in the model theory of modules over a ring R is the investigation of the pure-injective indecomposab]e modules over R, the set of (isomorphism types of) which may be given a quasi-compact topology forming the Ziegler spectrum Zg R of R. There are various natural *This paper was written during the visit of the second author to the University of Manchester supported by EPSRC Grant GR/L68827. He would like to thank the University for the kind hospitality. 268

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“…• • For many generalised Weyl algebras (in the sense of [3]), including the first Weyl algebra A 1 and its localisation B 1 , the quantum Weyl algebra A q (q = 0, 1) and the universal enveloping algebra U sl 2 , there is a wide poset of pointed finitely presented modules, hence if the field is countable, a superdecomposable pure-injective ([40, 7.5, §6], using a construction from [26]). …”

Section: 3])mentioning

confidence: 99%