Quantum differential operators on the quantum plane

The universal enveloping algebra of a Lie algebra acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl₂.     

Extending Sets of Idempotents to Ring Extensions

In this paper the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, R, a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete set of centrally primitive idempotents) can be constructed for an intrinsic extension, T, from a corresponding set in the base ring R. Examples and applications are given for rings that occur in functional analysis and group ring theory.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

Tangential Quantum Cohomology of Arbitrary Order

Kock has previously defined a tangency quantum product on formal power series with coefficients in the cohomology ring of any smooth projective variety, and thus a ring that generalizes the quantum cohomology ring. We further generalize Kock's construction by defining a dth-order contact product and establishing its associativity.     

Lattice theory for certain classes of rings

Conclusion We observe that every Boolean ring is aP ₁-ring, where each element is its own minimal idempotent duplicator; a Boolean ring with unity is also aP ₂-ring, where each element is the maximal idempotent annihilator of its complement; and thus,P ₁-rings andP ₂-rings are generalisations of a Boolean ring in terms of its lattice structure. Then, it is only natural that these two classes of rings have such close affiliations with the lattice structure of a Boolean ring.     

Kaluza-Klein pistons with noncommutative extra dimensions

We calculate the scalar Casimir energy and Casimir force for an ℝ³ × N Kaluza-Klein piston setup in which the extra-dimensional space N contains a noncommutative two-dimensional sphere S _fz. We study the cases with T ^d ×S _fz and S _fz as the extra-dimensional spaces, where T ^d is the d-dimensional commutative torus, and examine the validity of the results and the regularization obtained in the piston setup in each case. We examine the Casimir energy with one-loop corrections for one piston chamber due to the self-interacting scalar field in the noncommutative geometry. We compute with some approximations. We compare the obtained results with the results of analogous computations for the Minkowski space-time M ^D . In conclusion, we discuss the stabilization of the extra-dimensional space in the piston setup.     

On the Formation of Vortex Rings

The production of a vortex ring formed by using a piston to drive fluid through an orifice is considered. A cylindrical vortex sheet is supposed to be formed initially which rolls up into a vortex ring. Energy and momentum are conserved during rollup and determine the speed and size of the ring. It is shown that these quantities are independent of the vorticity distribution in the core of the ring. Reasonable agreement with experimental observations is found. A speculation is made about the criterion for the rings to be laminar or turbulent.     

Semilocal rings whose adjoint group is locally supersoluble

An associative ring R, not necessarily with an identity element, is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that if the adjoint group of a semilocal ring R is locally supersoluble, then R is locally Lie-supersoluble and its Jacobson radical is contained in a locally Lie-nilpotent ideal of finite index in R.     

Ring of normal cones

Normal categories were introduced by Nambooripad [4], who proved that the set of normal cones in such a category forms a regular semigroup. In this paper, we specialize normal categories to what we call RR-categories, and prove that the set of normal cones in such a category forms a regular ring. As an example, we analyze the regular ring of finite-rank operators on an infinite-dimensional vector space in terms of RR-categories. This example can be realised as the characterisation of a regular ring in terms of its left ideals.     

Irreducibility in the Total Ring of Quotients

Let R be a ring whose total ring of quotients Q is von Neumann regular. We investigate the structure of R when it admits an ideal that is irreducible as a submodule of the total ring of quotients. We characterize those rings which contain a maximal ideal that is irreducible in its total ring of quotients Q. An integral domain has a Q-irreducible ideal which is a maximal ideal if and only if R is a valuation domain. We show that when the total ring of quotients of R is von Neumann regular, then having a maximal ideal that is Q-irreducible is equivalently to certain valuation like properties, including the property that the regular ideals are linearly ordered.     

On Von Neumann Regular Rings of Skew Generalized Power Series

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S?{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].     

Semilocal rings with Engel conditions

The relation between the Engel structure of a semilocal ring and that of its multiplicative group is investigated. Suppose that every local ring whose multiplicative group satisfies an m-Engel condition for some positive integer m is an f (m)-Engel ring for some function f . It is proved that under this condition a corresponding statement holds for every semilocal ring which is generated by its multiplicative group. Received: 20 September 2005     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

Homomorphic images of subdirectly irreducible rings

We prove that every ring is a proper homomorphic image of some subdirectly irreducible ring. We also show that a finite ring R does not need to be isomorphic to the factor of a subdirectly irreducible ring by its monolith as well as R does not need to be a homomorphic image of a finite subdirectly irreducible ring. We provide an analogous characterization also for varieties of rings with unity, for the quasiregular rings, for the rings with involution and for their subvarieties of commutative rings.     

Injective and automorphism-invariant non-singular modules

Every automorphism-invariant non-singular right A-module is injective if and only if the factor ring of the ring A with respect to its right Goldie radical is a right strongly semiprime ring.     

Partial Crossed Products and Goldie Rings

In this paper we consider the transfer of the property of being a left Goldie ring between a ring A and its partial crossed product A*_α G by a twisted partial action α of a group G on A.     

Semilocal rings with <Emphasis Type="Italic">n</Emphasis>-Engel multiplicative group

An associative ring R with unit element is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that the multiplicative group R* of a semilocal ring R generated by R* satisfies an n-Engel condition for some positive integer n if and only if R is m-Engel as a Lie ring for some positive integer m depending only on n.Received: 21 January 2003     

The Structure of Finite Local Principal Ideal Rings

A ring R is called a principal ideal ring (PIR), if each ideal of R is a principal ideal. A local ring (R, 𝔪) is an artinian PIR if and only if its maximal ideal 𝔪 is principal and has finite nilpotency index. In this article, we determine the structure of a finite local PIR.     

The normalizer of the automorphism group of a module arising under extension of the base ring

Let Λ te an arbitrary associative ring with unity and let R be its unital subring contained in the center of Λ. Further, let M=λ M be a left free Λ-module of finite rank. In this paper, the normalizer of the subgroupAut(λM) of automorphisms of the module λM in the groupAut(_RM) of automorphisms of the module_RM is computed. If the ring Λ is additively generated by its invertible elements, then the above normalizer coincides with the semidirect product of the normal subgroupAut(λM) and a subgroup isomorphic to the groupAut(Λ/R) of all ring automorphisms of the ring Λ that are identical on R. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 133‐135. Translated by A. I. Skopin.     

Weakly left localizable rings

A new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S^?1R of the ring R. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals.