Torsion simple modules over the quantum spatial ageing algebra

Various classes of simple torsion modules are classified over the quantum spatial ageing algebra (this is a Noetherian algebra of Gelfand-Kirillov dimension 4). Explicit constructions of these modules are given and for each module its annihilator is found.     

Description of bilateral ideals in a class of noncommutative rings. II

For generalized Weyl algebras containing the universal enveloping algebra Usl (2,K) of the Lie algebra sl (2) over a field with characteristic zero, bilateral ideals are classified. We show that a product of ideals is commutative and any proper ideal can be uniquely decomposed into a product of primary ideals.     

Solutions to two questions about the Weyl algebras

Affirmative answers are given to the following two questions about the Weyl algebras: a question of J. Alev: Does the first Weyl algebra contain a non-noetherian subalgebra?, and a question of T. Lenagan: Is there a uniserial module of length over the Weyl algebra with a holonomic submodule such that is non-holonomic?

     

Criteria for a ring to have a left Noetherian left quotient ring

Two criteria are given for a ring to have a left Noetherian left quotient ring (to find a criterion was an open problem since 70's). It is proved that each such ring has only finitely many maximal left denominator sets.     

On approximations of the de Rham complex and their cohomology

For a commutative algebra R, its de Rham cohomology is an important invariant of R. In the paper, an infinite chain of de Rham-like complexes is introduced where the first member of the chain is the de Rham complex. The complexes are called approximations of the de Rham complex. Their cohomologies are found for polynomial rings and algebras of power series over a field of characteristic zero.     

The Largest Left Quotient Ring of a Ring

The left quotient ring (i.e., the left classical ring of fractions) Q_cl(R) of a ring R does not always exist and still, in general, there is no good understanding of the reason why this happens. In this article, existence of the largest left quotient ring Q_l(R) of an arbitrary ring R is proved, i.e., Q_l(R) = S₀(R)^?1R where S₀(R) is the largest left regular denominator set of R. It is proved that Q_l(Q_l(R)) = Q_l(R); the ring Q_l(R) is semisimple iff Q_cl(R) exists and is semisimple; moreover, if the ring Q_l(R) is left Artinian, then Q_cl(R) exists and Q_l(R) = Q_cl(R). The group of units Q_l(R)* of Q_l(R) is equal to the set {s^?1t | s, t ∈ S₀(R)} and S₀(R) = R ∩ Q_l(R)*. If there exists a finitely generated flat left R-module which is not projective, then Q_l(R) is not a semisimple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semiprime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators).     

Dixmier's Problem 6 for the Weyl Algebra (The Generic Type Problem)

ABSTRACT

In Dixmier (1968 Dixmier , J. ( 1968 ). Sur les algèbres de Weyl . Bull. Soc. Math. France 96 : 209 – 242 . [CSA] [Crossref] , [Google Scholar]), the author posed six problems for the Weyl algebra A ₁ over a field K of characteristic zero. Problems 3, 6, and 5 were solved respectively by Joseph (1975 Joseph , A. ( 1975 ). The Weyl algebra—semisimple and nilpotent elements . Amer. J. Math. 97 ( 3 ): 597 – 615 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and Bavula (2005a Bavula , V. V. ( 2005a ). Dixmier's Problem 5 for the Weyl algebra . J. Algebra 283 ( 2 ): 604 – 621 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). Problems 1, 2, and 4 are still open. In this article a short proof is given to Dixmier's problem 6 for the ring of differential operators 𝒟 (X) on a smooth irreducible algebraic curve X. It is proven that, for a given maximal commutative subalgebra C of 𝒟 (X), (almost) all noncentral elements of it have the same type, more precisely, have exactly one of the following types: (i) strongly nilpotent; (ii) weakly nilpotent; (iii) generic; (iv) generic, except for a subset K*a + K of strongly semi-simple elements; (iv) generic, except for a subset K*a + K of weakly semi-simple elements, where K* := K\{0}. The same results are true for other popular algebras.     

The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Let A_n be the nth Weyl algebra and P_m be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {A_n⊗P_m} is proved: an algebraAadmits a finite setδ₁,…,δ_sof commuting locally nilpotent derivations with generic kernels andiffA?A_n⊗P_mfor somenandmwith2n+m=s, and vice versa. The inversion formula for automorphisms of the algebra A_n⊗P_m (and for ) has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287-330]) given, then (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given, then. Any automorphism is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102-119], a similar result is proved for .One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242. [6]] problem 1) into a single question, (JD): is aK-algebra endomorphismσ:A_n⊗P_m→A_n⊗P_man algebra automorphism providedσ(P_m)⊆P_mand? (P_m=K[x₁,…,x_m]). It follows immediately from the inversion formula that this question has an affirmative answer iff both conjectures have (see below) [iff one of the conjectures has a positive answer (as follows from the recent papers [Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42(2) (2005) 435-452. [10]] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv:math.RA/0512171. [5]])].     

Some generalizations of the Eisenstein criterion

We obtain a generalized Eisenstein criterion for a certain class of -graded rings and for rings of skew polynomials A[x, ], where A is a commutative integral domain with unique factorization into prime factors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 983–985, July 1990.     

Left localizable rings and their characterizations

A new class of rings, the class of left localizable rings, is introduced. A ring R is left localizable if each nonzero element of R is invertible in some left localization $S^{?1}R$ of the ring R. Explicit criteria are given for a ring to be a left localizable ring provided the ring has only finitely many maximal left denominator sets (e.g., this is the case if a ring has a left Noetherian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets is a left localizable ring iff its left quotient ring is a direct product of finitely many division rings. A characterization is given of the class of rings that are finite direct product of left localization maximal rings.