Characterizations of some rings with <Emphasis Type="Italic">C</Emphasis>-projective, <Emphasis Type="Italic">C</Emphasis>-(FP)-injective and <Emphasis Type="Italic">C</Emphasis>-flat modules

Let R be a commutative ring and C a semidualizing R-module. We investigate the relations between C-flat modules and C-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.     

On generalized k-syzygy modules

In this paper, we first introduce the notion of generalized k-syzygy modules, and then give an equivalent characterization that the class of generalized k-syzygy modules coincides with that ofω-k-torsionfree modules. We further study the extension closure of the category consisting of generalized k-syzygy modules. Some known results are obtained as corollaries.     

Radicals of <Emphasis Type="Italic">l</Emphasis>-rings and special classes of <Emphasis Type="Italic">l</Emphasis>-modules

A special class of lattice-ordered modules is studied. We show that for any special class of l-modules we can define a special class of l-rings. The special radical of an l-ring R can be represented as the intersection of the l-annihilators of l-modules over R belonging to the special class. The prime radical of an l-ring R can be represented as the intersection of the l-annihilators of l-prime l-modules over R.     

Degenerations of <Emphasis Type="Italic">k</Emphasis>[<Emphasis Type="Italic">t</Emphasis>]/(<Emphasis Type="Italic">t</Emphasis><Superscript>r</Superscript>)-Modules Giving Stratification of the Orbits

Let Λ be a finitely generated associative k-algebra where k is an algebraically closed field. For each natural number d, we have the variety of d-dimensional module structures on k^d given by the multiplication of the elements from a generating set of Λ. The general linear group Gl_d(k) acts on this variety by conjugation and the orbits under this action correspond to isomorphism classes of d-dimensional Λ-modules. For two d-dimensional Λ-modules M and N one says that M degenerates to N if the orbit corresponding to N is in the Zariski-closure of the orbit corresponding to M. Now in this situation the stabilizers of the elements in the orbit corresponding to N acts on the orbit corresponding to M. In this paper we characterize degenerations of k[t]/(t^r)-modules with the property that for each y in the orbit corresponding to N, there is an x_y in the orbit corresponding to M such that the orbit corresponding to M is the disjoint union of orbits of the x_y’s under the action of the stabilizer of y where y runs through the orbit corresponding to N. Presented by Idun ReitenMathematics Subject Classifications (2000) 14L30, 16G10.     

Structure of <Emphasis Type="Italic">H</Emphasis>-Semiprime Artinian Algebras

Let H be a Hopf algebra over a field. It is proved that every H-semiprime right artinian left H-module algebra A is quasi-Frobenius and H-semisimple. If H grows slower than exponentially, then all H-equivariant A-modules are shown to be A-projective. With the additional assumption that H is cosemisimple it is proved that the Jacobson radical of any right artinian left H-module algebra is stable under the action of H.     

On <Emphasis Type="Italic">k</Emphasis>-Spaces and <Emphasis Type="Italic">k</Emphasis><Subscript><Emphasis Type="Italic">R</Emphasis></Subscript>-Spaces

In this note we study the relation between k _R -spaces and k-spaces and prove that a k _R -space with a σ-hereditarily closure-preserving k-network consisting of compact subsets is a k-space, and that a k _R -space with a point-countable k-network consisting of compact subsets need not be a k-space. This work was supported by the NSF of China (10271056).     

κ-Gorenstein Modules

Let A and F be artin algebras and ∧UГa paper, we first introduce the notion of k-Gorenstein faithfully balanced selforthogonal bimodule. In this modules with respect to ∧UГ and then characterize it in terms of the U-resolution dimension of some special injective modules and the property of the functors Ext^i （Ext^i （-, U）, U） preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of ∧UГwith finite left or right injective dimension.     

Tensor Products of Hilbert Modules Over Locally <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

In this paper the tensor products of Hilbert modules over locally C ^*-algebras are defined and their properties are studied. Thus we show that most of the basic properties of the tensor products of Hilbert C ^*-modules are also valid in the context of Hilbert modules over locally C ^*-algebras.     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript> and rings over which the class of finitely generated strongly gorenstein projective modules is closed under extensions

In this paper, we consider the rings over which the class of finitely generated strongly Gorenstein projective modules is closed under extensions (called fs-closed rings). We give a characterization about the Grothendieck groups of the category of the finitely generated strongly Gorenstein projective R-modules and the category of the finitely generated R-modules with finite strongly Gorenstein projective dimensions for any left Noetherian fs-closed ring R.     

Extensions of <Emphasis Type="Italic">GM</Emphasis>-rings

It is shown that a ring R is a GM-ring if and only if there exists a complete orthogonal set { e ₁,...,e _n } of idempotents such that all e _i Re _i are GM-rings. We also investigate GM-rings for Morita contexts, module extensions and power series rings.This work was supported by the Natural Science Foundation of Zhejiang Province.     

Simple <Emphasis Type="Italic">SL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-modules with normal closures of maximal torus orbits

Let T be the subgroup of diagonal matrices in the group SL(n). The aim of this paper is to find all finite-dimensional simple rational SL(n)-modules V with the following property: for each point v∈V the closure [`(Tv)]\overline{Tv} of its T-orbit is a normal affine variety. Moreover, for any SL(n)-module without this property a T-orbit with non-normal closure is constructed. The proof is purely combinatorial: it deals with the set of weights of simple SL(n)-modules. The saturation property is checked for each subset in the set of weights.     

Relative <Emphasis Type="Italic">FI</Emphasis>-injective and <Emphasis Type="Italic">FI</Emphasis>-flat modules

Let R be a ring, n a fixed nonnegative integer and FP _n (F _n ) the class of all left (right) R-modules of FP-injective (flat) dimensions at most n. A left R-module M (resp., right R-module F) is called n-FI-injective (resp., n-FI-flat) if Ext ¹(N,M) = 0 (resp., Tor ₁(F,N) = 0) for any N ∈ FP _n . It is shown that a left R-module M over any ring R is n-FI-injective if and only if M is a kernel of an FP _n -precover f: A → B with A injective. For a left coherent ring R, it is proven that a finitely presented right R-module M is n-FI-flat if and only if M is a cokernel of an F _n -preenvelope K → F of a right R-module K with F projective if and only if M ∈^⊥ F _n . These classes of modules are used to construct cotorsion theories and to characterize the global dimension of a ring.     

Liftable <Emphasis Type="Italic">D</Emphasis><Subscript>4</Subscript>-covers

Let k be an algebraically closed field of characteristic p and let G \hookrightarrow Aut_k(k[[ t ]] ){G \hookrightarrow {\rm Aut}_k(k\left[\kern-0.15em\left[ t \right]\kern-0.15em\right] )} be a faithful action on a local power series ring over k. Let R be a discrete valuation ring of characteristic 0 with residue field k. One asks, whether it is possible to find a faithful action G \hookrightarrow Aut_R(R[[ t ]] ){G \hookrightarrow {\rm Aut}_R(R\left[\kern-0.15em\left[ t \right]\kern-0.15em\right] )} which reduces to the given action, i.e., a lift to characteristic 0. We show that liftable actions exists in the case that G  =  D ₄ and p  =  2. In fact we introduce a family, the supersimple D ₄-actions, which can always be lifted to characteristic 0.     

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

On the evaluation at (<Emphasis Type="Italic">j</Emphasis>, <Emphasis Type="Italic">j</Emphasis><Superscript>2</Superscript>) of the Tutte polynomial of a ternary matroid

F. Jaeger has shown that up to a ± sign the evaluation at (j, j ²) of the Tutte polynomial of a ternary matroid can be expressed in terms of the dimension of the bicycle space of a representation over GF(3). We give a short algebraic proof of this result, which moreover yields the exact value of ±, a problem left open in Jaeger's paper. It follows that the computation of t(j, j ²) is of polynomial complexity for a ternary matroid.E. Gioan: C.N.R.S., MontpellierM. Las Vergnas: C.N.R.S., Paris     

Composite Integers n for Which φ（n）｜n- 1

In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.     

Polyhedral <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">2</Emphasis></Subscript>

 Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They coincide with the usual K-theoretic groups in the special case when the polytope is a unit simplex and can be thought of as compact/polytopal substitutes for the tame automorphism groups of polynomial algebras. Relative to the classical case, many new aspects have to be taken into account. We describe these groups explicitly when the underlying polytope is 2-dimensional. Already this low-dimensional case provides interesting classes of groups. Received: 13 December 2001 / Revised version: 24 June 2002 The second author was supported by the Deutsche Forschungsgemeinschaft, INTAS grant 99-00817 and TMR grant ERB FMRX CT-97-0107 Mathematics Subject Classification (2000): 14L27, 14M25, 19C09, 52B20     

An Algorithm for Corona Solutions on <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> (<Emphasis Type="Italic">D</Emphasis>)

We give an algorithm to find corona solutions in H ^∞ (D) for polynomial input data. Partially supported by NSF Grant DMS-0400307.     

The quantum double of the Yangian of the Lie superalgebra <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">m,n</Emphasis>) and computation of the universal <Emphasis Type="Italic">R</Emphasis>-matrix

The Yangian double DY(A(m, n)) of the Lie superalgebra A(m, n) is described in terms of generators and defining relations. We prove the triangular decomposition for Yangian Y(A(m, n)) and its quantum double DY(A(m, n)) as a corollary of the PBW theorem. We introduce normally ordered bases in the Yangian and its dual Hopf superalgebra in the quantum double. We calculate the pairing formulas between the elements of these bases. We obtain the formula for the universal R-matrix of the Yangian double. The formula for the universal R-matrix of the Yangian, which was introduced by V. Drinfel’d, is also obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 185–208, 2005.