Lecture Notes ni Mathematics Edited by .A Dold and .B Eckmann 8231 .J L. Bueso R Jara B. Torrecillas (Eds.) gniR yroehT Proceedings of a Conference held ni Granada, Spain, Sept. 1-6, 1986 Springer-Verlag Berlin Heidelberg NewYork London Paris oykoT Editors Jose Luis Bueso Pascuat Jara Bias Torrecillas Departamento de Algebra, Facultad de Ciencias Universidad de Granada 18071 Granada, Spain Mathematics Subject Classification (1980): 16-02, 16-06, 16A 03, 16A06, 16 A08, 16A26, 16A30, 16A54, 16A55, 16A61, 16A63 ISBN 3-540-19474-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19474-6 Springer-Vertag New York Berlin Heidelberg This work subject is to copyright. whether All reserved, rights are the whole or part the of material is concerned, specifically the rights of translation, reprinting, of re-use illustrations, recitation, broadcasting, reproduction no microfilms or ni storage ways, and other ni banks. data Duplication of this publication or the permitted provisions thereof parts only under is of the German Copyright Law of September ,9 1965, ni 24, version of June its a copyright and 1985, fee always must be paid. fall the Violations under prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1988 Printed ni Germany and binding: Printing Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 EDITORIAL These proceedings contain papers presented in the meeting in "Ring Theory" celebrated in Granada (SPAIN). Not all the lectures and communi- cations appear here, some of them were given by invited speakers, who were unable to attend, and others are being published elsewhere. The editors want to thank to the participants for their cooperation during the meeting and in preparing this volume; especially we thank to Professor F. van Oystaeyen. VI A C K N O W L E D G E M E N T The meeting held at the University of Granada from September ist until September 6th, is financed by the following corporations: - Junta de Andalucfa. - CAYCIT. - Excelentisima Diputaci6n Provincial de Granada. - Banco Exterior de Espa~a. - Caja General de Ahorros de Granada. - Proyecto de investigaci6n CAYCIT: "Teoria de anillos". - Facultad de Ciencias. - Universidad de Granada. We thank the staff of the Faculty of Sciences and Department of Algebra for providing facilities and support in the organization of this meeting. TABLE OF CONTENTS ARA, P., "Stable range of alef-nought-continuous regular ring". AVAMI-VAN OYSTAEYEN. On filtered rings with noetherian associated graded rings. 8 BEATTIE, N., "Duality theorems for group actions and gradings". 28 BRUNGS, H.H. "Chain rings and valuations". 33 BUSQUE, C., "Directly finite aleph-nought complete regular rings are unit-regular". 38 CAENEPEEL, S., "Cancelations theorems for Projective Graded Modules" 50 CAUCHON, G. "Centraliseurs dans les anneaux de polyn~mes diff6rentiels formels et leurs corps de fractions". 60 CEDO, F., "Regular group algebras whose maximal right and left quotient rings coincide". 69 DICKS, W., "A survey of recent work on the cobomology of one-relator associative algebras". 75 ESSANNOUNI, H.; KAIDI. "Semiprime alternative rings with ascending condition". 82 GARCIA HERNANDEZ, J.L., "Continuous and PF-rings of quotients" 94 GOMEZ PARDO, J.LO., "Rings of quotients of endomorphism rings". 106 HERMIDA, SANCHEZ GIRALDA., "Some criteria for solvability of systems of linear equations over modules". 122 LE BRUYN, L., "Center of generic division algebras and zeta-functions". 135 LORENZ, M., "Frobenius reciprocity and G of skew group rings". 165 o MALLIAVIN, M.P., "Alg~bre homologique et op~rateurs diff~rentiels". 173 MENAL, P., "Cancellation modules over regular rings". 187 OKNINSKI, J., "Noetherian property for semigroup rings". 209 PEREZ ESTEBAN, D., "Semirings and spectral spaces". 219 ROGGENKAMP, K.W.,SCOTT,L."Some new progress on the isomorphism problem for integral group ring". 227 SANDLING, R., "A proof of the class sum correspondence using the real group algebra". 237 Vl SANGHARE, M.; KAIDI., "Une caracterlzatzon des anneaux artiniens h ideaux principaux". 245 SAORIN, M., "Krull and Gabriel dimension relative to a linear Topology" 255 SUSPERREGUI, J., "On determinantal ideals over certain non commutative rings". 269 TEPLY, M.L., "Large subdirect products" 283 TORRECILLAS, B., "Socle and semicocritical series". 305 VERSCHOREN, A., "Local Cohomology of non commutative rings: a geometric approach". 316 PARTI CI PANTS. .A Alvarez Dot~, Departamento de Algebra, Faculta~ de MatemAticas y Quimicas, 30001 Murcia, Espa~a. S. .A Amitsu{, Department of Mathematics, Hebrew University, 5erusalem, Israel. P. Ara, Departamento de Algebra, Universidad Aut6noma de Barcelona, Bellaterra, Barcelona, Espa~a. 5. Asensio, Departamento de Algebra, Facultad de Matem~ticas y Qufmicas, 30001 Murcia, Espa~a. M. .J Asensio del Aguila, Departamento de Algebra, Universidad de Granada, 18071 Granada, Espa~a. G. Baccella, Instituto Matematico, Universit~ del'Aquila, Via Roma, 33, @TIO0 L'Aquila, Italy. J. Barja, Departamento de Algebra, Universidad de Santiago de Compostela, La Coru~a, Espa~a. 3. M. Barja, Departamento de Algebra y Fundamentos, Facultad de Ciencias. Universidad de M~laga, Apto. 5g, 2gOOO-M~laga. Espa~a. M. Beattie, Department of Mathematics, Mount Saint Vincent University, I@@ Bedford Highway, Halifax, Nova Scotia, Canada B3M 2J6. S. Brenner,Department of Pure Mathematics, University of Liverpool, P.O. Box i47, Liverpool, U.K. L 6g 3BX H.H. Brungs, Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada. J. L. Bueso, Departamento de Algebra, Universidad de Granada, 18071 Granada, Espa~a. W.D. Burgess, Dept. of Mathematics, University of Ottawa, Ottawa, Canada KIN 6NI. .C Busqu~ Roca, Departamento de Matematicas, Universidad Autonoma de Barcelona, Bellaterra, Barcelona, Espa~a. M.C.R. Butler, Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool, U.K. L @g 3BX S, Caenepeel, University of Brussels, VUB, Fac, Applied Sciences, Pleinlann 2, B-lOS0, Brussel, Belgium. I. Calais, .U E. .R Sciences de Reims, Departement de Math~matiques, Moulin de la Housse, B. P. 347, Reims Cedex, France. L. Carini, Dipartimento di Matematica dell'Universita, Via C. Battisti .N go,g8100 Messina, Italy. G. Cauchon, Departement de Mathematiques, U.F.R. des Sciences de Reims, Moulin de la Housse, 51052 Reims Cedex, France. F. Ced6, Departamento de Algebra, Universidad Aut6noma de Barcelona, Bellaterra, Barcelona, Espa~a. J. A. Clua Sampietro, Departamento de Matematiques, Universidad Aut6noma de Barcelona, Bellaterra, Barcelona, Espa~a. J. R. Delgado P~rez. Departamento de Algebra y Fundamentos, Facultad de Matem&ticas, Universidad Complutense, 28040 Madrid, Espa~a. A. Del Rio,Departamento de Algebra, Facultad de Matem&ticas y Quimicas, 30001 Murcia, Espa~a. W. Dicks, Departamento de Algebra, Universidad Autonoma de Barcelona, Bellaterra, Barcelona, Espa~a. L. Espa~ol Gonz~lez, Colegio Universitario de la ~oija, Obispo Bustamante 3, 2@001 Logro~o, Espa~a. H. Essannouni, Department de Math~matiques, Facult~ des Sciences, VIII Rabat, B. P. 1014, Maroc .A Facchini, Instituto di Matematica, Informatica e Sistemistica, Via Zanon 8, 33100 Udine, Italy. A. FernAndez, Departamento de Algebra y Fundamentos, Facultad de Ci enci as. Uvneit si dad de MAIaga, Apt o. 5g, 2~080-MAI aga. Espa~a. E. For manek. Mathematics Department, Pennsyl vani a State University, University Park, PA 16802, U.S.A. .J L. Garcia Her n~ndez, Depar tamento de Algebra, Facul tad de MatemAticas y Quimicas, 30001 Murcia, Espa~a. J.L. Gomez Pardo, Departamento de Algebra, Facultad de Matem~ticas y. Quimicas, :90001 Murcia, Espa~a. S. Gonz~l ez, Departamento de Algebra, Facul tad de CI encias, Universidad de 7aragoza, 5000g Zaragoza, Espa~a. M. Hacque, Institut de Math~matique et Informatique de L'I.S.M., Universit~ Claude Bernard- Lyon I. 43, boulevard du ii Novembre iglB, 69822 Villeurbanne Cedex, France. M. D. HerberA, Departamento de Matematicas, Universidad Aut6noma de Barcelona, Bellaterra, Barcelona, Espa~a. J. A. Her mi da, Depar tamento de Algebra, Facultad de Ci enci as, Universidad de Valladolid, 47005 Valladolid, Espa~a. .M Her v~, Depar tement de Mathematiques, U.F.R. des Sciences de Reims, Moulin de la Housse, 510@a Reims Cedex, France. P. /ara Martinez, Departamento de Algebra, Universidad de Granada, 18071 Granada, Espa~a. Kaidi E1 Amine, Department de k~ath@matiques, Facult@ des Sciences, Rabat, .B P. 1014, Maroc. .A Kupf er ot h, Mat hemati sches I nsti rut, Uni vet si tit D~ssel doff, Universit~tsstraBe i, D-4000 Ddsseldorf, West Germany. L. Le Bruyn, Dept. Mathematics, . I U. .A Uni versiteitsplein , i B-2810 Wilryk, Belgium. .A Leroy, Universit~ de l'Etat ~ Mons, 15 Avenue Maistrion, 7000 Mons, Belgique. .F Loonstra, Den Haag, Haviklaan 25, Holland. .M Lorenz, Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois @0115-2888, U.S.A. M.P. Malliavin, Institut Henri Poincare, II Rue Pierre et Marie Curie, ?5005, Paris, France. .S Mamadou, Department de Math~matiques, Facutl@ des Sciences, Rabat, B. P. I014, Maroc. .M Martin, Departamento de Algebra, Universidad de Granada, IBO?l Granada, Espa~a. .C Mar tl nez, Depar tament o de A1 gebr a, Facul tad de Ci enci as, Universidad de Zaragoza, 50OOg Zaragoza, Espa~a. P. Menal , Depar tamento de A1 gebr a, Uni versi dad Autonoma de Barcelona, Bellaterra, Barcelona, Espana. .P Misso, Dipartamento di Matematicas, UniversiSa di Palermo, Via Archifari 34, 90100 Palermo, Italy. .J Moncasi Sol sona, Deparatm ento de Mat emati ques, Uni versi dad Aut6noma de Barcelona, Bellaterra, Barcelona, Espa~a. .M Nordin, De Valk ~5, B-2410 Mortiel, Belgium. J. Okni nski , I nsti tute of Mathematics, Uni versity of Warsaw, O0-gOl War saw, PKiN, Poland. C. Or do~ez Canada, Depar tament o de Al gebr a, Uni vet si dad de Granada, 18071 Granada, Espa~a. D. P@rez Esteban, C.E.C.I.M.E. , ~errano 123, 2800B Madrid, Espa~a. .I M. Pi acenti ni Cat taneo, Department of Mathematics, Second University of Rome, Via Orazio Raimondo, 00173 Roma, Italy XI 3. Raynaud, Institut de Mathematique eL Informatique de L'I.S.~4., Universite Claude Bernard- Lyon I. 43, boulevard du ii Novembr e I gl 8, 89622 Vi 11 eur banne Cedex, France. K.W. Roggenkamp, Math. I nsti tut B/3, Uni ver si tat Stuttgart, Pfaffenwaldring ST, 7000 Stuttgart 80, West-Germany. R. Sandllng, Department of Mathematics, The University, Manchester MI3 gPl, England. .M Saorin,Departamento de Algebra, Facultad de Matem&ti cas y Quimicas, 30001 Murcia, Espa~a. J. Susper r egui , Depar tamento de Matem~ti cas, Facul tad de Informatica, Apdo 84g, San ~bastian, 20080 Espa~a. J. Tena Ayuso, Departamento de Algebra y ~ometria, Facultad de Ciencias, Uni ver si dad de Val i adoli d, 47005 Val I adolid, Espa~a. M. .L Teply, Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin ~320i, U.S.A. B. Torrecillas, Departamento de Algebra, Universidad de Granada, 18071 Granada, Espa~a. F. Van Oy~taeyen, Dept. Mathematics, U.I.A. Universiteitsplein I, B-2BIO Wilryk, Belgium. .A Verschoren, Faculteit der Wetenschapen, R.U.C.A. , Middelheimlaan, Antwerpen, Belgium P. Wauters, Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B, B-3030, Leuven, Belgium. R. Wi sbauer, Mathemati sches I nsti tut, Uni vet si t~t E~dss el dot f, Universit~tsstraBe I, D-4000 Ddsseldorf, West Germany. STABLE RANGE OF ALEPH-NOUGHT-CONTINUOUS REGULAR RINGS PERE ARA Departament de Matem~tiques, Universitat Aut~noma de Barcelona, Bellaterra, Barcelona, Spain. ABSTRACT. In this paper we show that if R iS a right Mo-continuous reg~ar ring, then the s~t of possible values for the stable range of R, sr(R), /S {1,2,=}. F~rther, sr (R) = 1 if and only if R iS di~ec2ly finite, and sr (R) <~ 2 if dna only if R iS an Hermite ring. All rings considered in this paper are associative with i, and all modules are unital. A ring R is said to be regular if for every a E R there exists an element b E R such that a = aba. Let R be any ring. A n-row x=(xl,...,x n) ~ n R is unimodular if XlR + ... + XnR = R, and x is reducible if there exist yl,...,Yn_l E R such that the (n - l)-row (x I + XnY 1 .... ,Xn_ 1 + XnYn_ )I is unimodular. R is said to have stable range n, sr(R) = n, if n is the least positive integer such that every unimodular (n+l)-row is reducible; the stable range of R is ~ if there does not exist any positive integer with this property. Recall [4, p. 465] that a ring R is right (left) Hermite if every Ix2 (2xl) matrix admits diagonal reduction. In other words, This work was partially supported by CAICYT grant 3556/83.