Local Morphisms and Modules with a Semilocal Endomorphism Ring

We study local morphisms in the setting of general noncommutative rings. In particular, we apply local morphisms to study endomorphism rings of modules. We use our constructions to determine classes of modules with semilocal endomorphism rings. For instance, we prove that every finitely presented right module over a semilocal ring has a semilocal endomorphism ring.Alberto Facchini was partially supported by Gruppo Nazionale Strutture Algebriche e Geometriche e loro Applicazioni of Istituto Nazionale di Alta Matematica, Italy, and by Università di Padova (Progetto di Ateneo CDPA048343 “Decomposition and tilting theory in modules, derived and cluster categories”).Dolors Herbera was partially supported by the DGI and the European Regional Development Fund, jointly, through Project BFM2002-01390, and by the Comissionat per Universitats i Recerca of the Generalitat de Catalunya.     

Strongly regular rings and rational identities of division rings

We look at division rings in the variety of strongly regular rings and show a connection to the study of rational identities on division rings.Research partially supported by a Grant from NSERC.     

Filtrations, mixed complexes, and cyclic homology in mixed characteristic

We describe an approach to computing cyclic homology suited to ground rings of characteristicp or mixed characteristic. We illustrate the method by computing HC for coordinate rings of affine plane curves and for maximal orders of semisimple algebras.Partially supported by NSF Grant No. DMS-8807203 and NSA Grant No. MDA 904-92-H-3026.     

Rigid Dualizing Complexes Over Commutative Rings

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211–3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes. This research was supported by the US–Israel Binational Science Foundation. The second author was partially supported by the US National Science Foundation.     

An algebraic dependence over the quasi-centre

Summary The authors study a class of rings generalizing both ξ-rings and rings which are (1) quasi-radical over their normalizer. The results are quite complete in the Noetherian case. In application several conditions on rings are shown to force the right duo condition. In particular it is shown that ξ-rings, algebras over the rationals satisfying (1), and algebraic algebras having no non central nilpotent elements (or a more general condition) are right duo or duo rings. Entrata in Redazione il 28 luglio 1972. This research has been supported by grant A7876 of the National Research Council of Canada. This research has been supported by grant A7877 of the National Research Council of Canada.     

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

On Polynomials Over Rickart Rings and Their Generalizations I

This paper continues the investigation of polynomials and formal power series over a ring with various annihilator conditions which were originally used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebras. Results of Armendariz on polynomial rings over a PP ring are extended to analogous annihilator conditions in nearrings of polynomials and nearrings of formal power series. These results are somewhat striking since, in contrast to the polynomial ring case, the nearring of polynomials or formal power series has substitution for its “multiplication” operation. These investigations provide an alternative viewpoint in illustrating the structure of polynomials and formal power series. Extensions of Rickart rings to formal power series rings are also discussed. The author was partially supported by the National Science Council, Taiwan under the grant number NSC 93-2115-M-143-001.     

SymplecticK 2 of Laurent polynomials,associated Kac-Moody groups and Witt rings

The first author was partially supported by the Alexander von Humboldt-foundation in 1988–89. The authors want to thank Professor A. Bak for his valuable advice onK ₂Sp(F), Witt rings and their relation.     

Rings over which all torsion theories cogenerated by simple modules are jansian

Summary It is given a caracterization of (associative) rings (with unit element) over which all torsion theories cogenerated by simple modules are jansian. In the commutative case it is shown that such rings are finite products of local rings.

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Riassunto Vengono caratterizzati gli anelli (associativi con unità) per i quali tutte le teorie di torsione cogenerate da moduli semplici sono Jansiane. Nel caso commutativo si prova che tali anelli sono prodotti finiti di anelli locali.

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This work was written while the second author was visiting Università dell'Aquila, supported by a grant from the Consiglio Nazionale delle Ricerche.     

Cafiero and Brooks–Jewett theorems for Vitali spaces

We introduce a common generalization of Boolean rings and lattice ordered groups called Vitali spaces and we give a version of Cafiero and Brooks–Jewett convergence Theorems for additive functions defined in a Vitali space with values in a Hausdorff commutative topological group. This article was supported by MURST, project “Analisi Reale” and by GNAMPA of Istituto Nazionale di Alta Matematica.     

Extensions and Pullbacks in QB-Rings

We prove a new extension result for QB-rings that allows us to examine extensions of rings where the ideal is purely infinite and simple. We then use this result to explore various constructions that provide new examples of QB-rings. More concretely, we show that a surjective pullback of two QB-rings is usually again a QB-ring. Specializing to the case of an extension of a semi-prime ideal I of a unital ring R, the pullback setting leads naturally to the study of rings whose multiplier rings are QB-rings. For a wide class of regular rings, we give necessary and sufficient conditions for their multiplier rings to be QB-rings. Our analysis is based on the study of extensions and the use of nonstable K-theoretical techniques. Presented by S. MontgomeryMathematics Subject Classification (2000) 16D30.Gert K. Pedersen: Deceased 15 March 2004.Partially supported by the DGI and European Regional Development Fund, jointly, through Project BFM 2002-01390, the Comissionat per Universitats i Recerca de la Generalitat de Catalunya and the Danish Research Council.     

Canonical maps to twisted rings

If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring , which is surjective in large degree. This result is a key step in the study of projectively simple rings. The proof relies on some results concerning the growth of graded rings which are of independent interest. D. Rogalski was partially supported by NSF grant DMS-0202479. J. J. Zhang was partially supported by NSF grant DMS-0245420 and Leverhulme Research Interchange Grant F/00158/X (UK).     

Units in alternative loop rings

In this paper, we show that certain well known theorems concerning units in integral group rings hold more generally for integral loop rings which are alternative. This research was supported in part by the Natural Sciences and Engineering Research Council, Grants No. A9087 and A8775.     

Generalized Stable Exchange Rings

In this paper, we investigate diagonal reductions of matrices over generalized stable exchange rings. We show that every regular matirx over generalized stable exchange rings with stable range 2 admits power diagonal reduction.AMS Subject Classification (1991): 16A30 16E50This work was supported by the National Natural Science Foundation of China.     

Morava K -theory rings for the modular groups in Chern classes

Morava K-theory rings of classifying spaces of the modular and quasi-dihedral groups are calculated in terms of Chern characteristic classes and the Honda formal group law. The author was supported by the INTAS 03-51-3251 and GRDF GEM1-3330-TB-03 grants.     

Separative cancellation for projective modules over exchange rings

A separative ring is one whose finitely generated projective modules satisfy the propertyA⊕A⋟A⊕B⋟B⊕B⇒A⋟B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ringR has an idealI withI andR/I both separative, thenR is separative. The research of the first and fourth authors was partially supported by a grant from the DGICYT (Spain) and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. That of the second author was partially supported by a grant from the NSF (USA). The final version of this paper was prepared while he was visiting the Centre de Recerca Matemàtica, Institut d'Estudis Catalans in Barcelona, and he thanks the CRM for its hospitality.     

On the optimum design of piston rings

We study a family of control problems that arise in the design of dynamically loaded bearings which have a minimum power loss; in particular, we study the design of piston rings with minimum parasitic power loss.This research was partially supported by NSF Grant No. MCS-77-05624.     

Additive endomorphisms of rings

We characterize finite rings all of whose additive endomorphisms are multiplicative also.This work was supported by Alabama State University Faculty Research Grant R-78-6.     

The Semigroup Structure of Left Clifford Semirings

In this paper,we generalize Clifford semirings to left Clifford semirings by means of the so-called band semirings.We also discuss a special case of this kind of semirings,that is, strong distributive lattices of left rings.     

Computing combinatorial decompositions of rings

Using Buchberger's Gröbner basis theory, we obtain explicit algorithms for computing Stanley decompositions, Rees decompositions and Hironaka decompositions of commutative Noetherian rings. These decompositions are of considerable importance in combinatorics, in particular in the theory of Cohen-Macaulay complexes. We discuss several applications of our methods, including a new algorithm for finding primary and secondary invariants of finite group actions on polynomial rings.Research supported by the Institute for Mathematics and its Applications, Minneapolis, with funds provided by the National Science Foundation