Artinian Rings and the H-Condition

It is proved that a ring R is right Artinian if and only if,for each countably generated right R-module M, there existsa finite subset F of M such that the annihilator of M in R equalsthe annihilator of F in R. 2000 Mathematics Subject Classification16P20.     

Tilting Cotorsion Pairs

Let R be a ring and T a 1-tilting right R-module. Then T isof countable type. Moreover, T is of finite type in the casewhere R is a Prüfer domain. 2000 Mathematics Subject Classification16D90, 16D30, 13G05 (primary), 03E75, 20K40, 16G99 (secondary).     

Noetherian Semigroup Algebras

A complete structural characterization of submonoids S of apolycyclic-by-finite group such that the semigroup algebra K[S]over a field K is right noetherian is obtained. It follows thatsuch algebras are also left noetherian. 2000 Mathematics SubjectClassification 16P40, 16S36, 20M25 (primary), 20F22, 20C07,20M10 (secondary).     

The Extension Group of the Simple Modules Over the First Weyl Algebra

The aim of this paper is to prove that for certain generalizedWeyl algebras A (including the first Weyl algebra A₁ over afield of characteristic zero) and for every simple left (right)A-module M, there are infinitely many non-isomorphic simpleleft (right) A-modules {N_i} such that (respectively ). 1991 Mathematics Subject Classification 18G15, 16G.     

Indecomposable flat cotorsion modules

An additive functor from the category of flat right R-modulesto the category of abelian groups is continuous if it is isomorphicto a functor of the form–_R M, where M is a left R-module.It is shown that for any simple subfunctor A of– M thereis a unique indecomposable flat cotorsion module U_R for whichA(U)0. It is also proved that every subfunctor of a continuousfunctor contains a simple subfunctor. This implies that everyflat right R-module may be purely embedded into a product ofindecomposable flat cotorsion modules. If CE(R) is the cotorsion envelope of R_R and S= End;_R CE(R),then a local ring monomorphism is constructed from R/J(R) toS/J(S). This local morphism of rings is used to associate asemiperfect ring to any semilocal ring. It also proved thatif R is a semilocal ring and M a simple left R-module, thenthe functor–_R M on the category of flat right R-modulesis uniform, and therefore contains a unique simple subfunctor.     

Towers of MU-Algebras and the Generalized Hopkins-Miller theorem

Our results are of three types. First, we describe a generalprocedure of adjoining polynomial variables to A-ring spectrawhose coefficient rings satisfy certain restrictions. A hostof examples of such spectra is provided by killing a regularideal in the coefficient ring of MU, the complex cobordism spectrum.Second, we show that the algebraic procedure of adjoining rootsof unity carries over in the topological context for such spectra.Third, we use the developed technology to compute the homotopytypes of spaces of strictly multiplicative maps between suitableK(n)-localizations of such spectra. This generalizes the famousHopkins–Miller theorem and gives strengthened versionsof various splitting theorems. 2000 Mathematics Subject Classification55N20, 55S35, 55T25 (primary), 16E40, 13D10 (secondary).     

Serial Rings with T-Nilpotent Prime Radical

In this paper we consider serial rings with T-nilpotent prime radical, factor-rings of which by the prime radical are right Noetherian rings. We prove that the prime quiver of such a ring is a disconnected union of cycles and chains. In the case when the prime quiver of such a serial ring is a chain the prime radical is nilpotent. For serial rings with nilpotent prime radical we introduce an analogue of Kupisch series. Presented by Yu. Drozd Mathematics Subject Classifications (2000) 16P40, 16G10.     

Constructions of Partial Difference Sets and Relative Difference Sets on p-Groups

By using some finite local rings, we construct some new partialdifference sets and relative difference sets on pgroups wherep is any prime. When p = 2, some of partial difference setsconstructed are reversible difference sets which include Dillon'sdifference sets.     

Localization in Hermitian K-Theory of Rings

Localization and dévissage theorems are proved for thehermitian K-theory of rings that are analogous to well-knowntheorems in algebraic K-theory. The proofs rely on, among otherthings, a study of derived categories, a generalization of atheorem of Pedersen and Weibel to the hermitian setting, anda cofinality result for triangular Witt groups. Applicationsinclude a proof of a conjecture of Karoubi and algebraic Bottperiodicity.     

Finite Modules of Finite Injective Dimension Over a Noetherian Algebra

Let R be a commutative Noetherian ring. Let P(R) (respectively,I(R)) be the category of all finite R-modules of finite projective(respectively, injective) dimension. Sharp [9] constructed acategory equivalence between I(R) and P(R) for certain Cohen–Macaulaylocal rings R. Thus many properties about finite modules offinite projective dimension can be connected with those of finiteinjective dimension through this equivalence.     

Elliptic Genera of Level N and Elliptic Cohomology

Elliptic genera of level N have been defined by F. Hirzebruch,generalising the earlier notion of elliptic genus due to S.Ochanine. We show that there are corresponding elliptic cohomologytheories which are naturally associated to such genera and thatthese are obtained from the level 1 case by algebraic extensionof the coefficient rings from level 1 to level N modular forms.     

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categoriesis presented. To each locally coherent Grothendieck categoryC a topological space, the Ziegler spectrum of C, is associated.It is proved that the open subsets of the Ziegler spectrum ofC are in bijective correspondence with the Serre subcategoriesof coh C the subcategory of coherent objects of C. This is aNullstellensatz for locally coherent Grothendieck categories.If R is a ring, there is a canonical locally coherent Grothendieckcategory RC (respectively, CR) used for the study of left (respectively,right) R-modules. This category contains the category of R-modulesand its Ziegler spectrum is quasi-compact, a property used toconstruct large (not finitely generated) indecomposable modulesover an artin algebra. Two kinds of examples of locally coherentGrothendieck categories are given: the abstract category theoreticexamples arising from torsion and localization and the examplesthat arise from particular modules over the ring R. The dualitybetween coh-(RC) and coh-CR is shown to give an isomorphismbetween the topologies of the left and right Ziegler spectraof a ring R. The Nullstellensatz is used to give a proof ofthe result of Crawley-Boevey that every character : K₀(coh-C) Z is uniquely expressible as a Z-linear combination of irreduciblecharacters. 1991 Mathematics Subject Classification: 16D90,18E15.     

On Totally Ordered Rings Whose Positive Cones are Finitely Generated

This paper is mainly dealt with the structure of totally ordered rings (t.o. rings) whose positive cones are finitely generated as multiplicative right ideals.AMS Subject Classification (1991): 16W80, 13J25     

The Green Correspondence for Infinitely Generated Modules

If G is a finite group, then the usual version of the Greencorrespondence applies to finitely generated kG-modules whenk is a field of characteristic p > 0 or a p-adic ring. Thepaper presents a categorical version of the Green correspondenceand a version of the Burry–Carlson–Puig theoremthat remain valid for arbitrary modules and coefficient rings.In this generality, however, it is not clear whether the Greencorrespondent of a finitely generated module is always finitelygenerated.     

关于内射模的P-平坦性

通过引用P-平坦模的定义,引入了右IPF环的概念,推广了右IF环的概念,这对研究IF环及QF环具有重要的作用,同时对右IPF环的性质作了一些刻画,得到了右IPF环的若干个等价命题;最后,用P-平坦模及右IPF环推出了正则环的一些等价条件．     

Hopf-Cyclic Homology and Relative Cyclic Homology of Hopf-Galois Extensions

Let H be a Hopf algebra and let M_s (H) be the category of allleft H-modules and right H-comodules satisfying appropriatecompatibility relations. An object in M_s (H) will be calleda stable anti-Yetter–Drinfeld module (over H) or a SAYDmodule, for short. To each M M_s (H) we associate, in a functorialway, a cyclic object Z_* (H, M). We show that our constructioncan be used to compute the cyclic homology of the underlyingalgebra structure of H and the relative cyclic homology of H-Galoisextensions. Let K be a Hopf subalgebra of H. For an arbitrary M M_s (K)we define a right H-comodule structure on so that becomes an object in M_s (H). Under some assumptions on K and M we computethe cyclic homology of . As a direct application of this result, we describe the relativecyclic homology of strongly graded algebras. In particular,we calculate the cyclic homology of group algebras and quantumtori. Finally, when H is the enveloping algebra of a Lie algebra g,we construct a spectral sequence that converges to the cyclichomology of H with coefficients in a given SAYD module M. Wealso show that the cyclic homology of almost symmetric algebrasis isomorphic to the cyclic homology of H with coefficientsin a certain SAYD module. 2000 Mathematics Subject Classification16E40 (primary), 16W30 (secondary).     

Nilpotent right alternative rings and bol circle loops

We determine the nilpotent right alternative rings of prime power oirder pⁿ n ≥ 4, which are not left alternative. Those which are strongly right alternative become Bol loops under the circle operation. The smallest Bol circle loop has order 16. There are six such loops, all of which appear to be new.     

Optimal backward perturbation bounds for the linear least-squares problem with multiple right-hand sides

Let A be an mn matrix, B be an mr matrix, and $$\stackrel{\¯}{X}$$be an approximate solution to the problem of minimizing ||AX–B||_F.In this note we consider the following open problem: find anexplicit expression of the optimal backward perturbation bound_F$${\eta }_{F}\left(\tilde{X}\right)$$ dehned by where is a positive number. This problem is solved when $$\tilde{X}$$is of full column rank.     

A New Look at the Burnside-Schur Theorem

The famous Burnside–Schur theorem states that every primitivefinite permutation group containing a regular cyclic subgroupis either 2-transitive or isomorphic to a subgroup of a 1-dimensionalaffine group of prime degree. It is known that this theoremcan be expressed as a statement on Schur rings over a finitecyclic group. Generalizing the latter, Schur rings are introducedover a finite commutative ring, and an analogue of this statementis proved for them. Also, the finite local commutative ringsare characterized in permutation group terms. 2000 MathematicsSubject Classification 20B10, 20B15, 05E99.