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The Hopkins-Levitzki Theorem

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Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from con- ferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be sent to the Mathematics Editor at either Springer Basel AG Birkhäuser P.O. Box 133 CH-4010 Basel Switzerland or Birkhauser Boston 233 Spring Street New York, NY 10013 USA Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TeX is acceptable, but the entire collection of files must be in one particular dialect of TeX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. The total number of pages should not exceed 350. The first-mentioned author of each article will receive 25 free offprints. To the participants of the congress the book will be offered at a special rate. Ring and Module Theory Toma Albu Gary F. Birkenmeier Ali Erdoğan Adnan Tercan Editors Birkhäuser Editors: Toma Albu Ali Erdoğan “Simion Stoilow” Institute of Mathematics Department of Mathematics of the Romanian Academy Hacettepe University P.O. Box 1-764 Beytepe Çankaya 010145 Bucharest 1 06532 Ankara Romania Turkey e-mail: [email protected] e-mail: [email protected] [email protected] Adnan Tercan Gary F. Birkenmeier Department of Mathematics Department of Mathematics Hacettepe University University of Louisiana at Lafayette Beytepe Çankaya Lafayette, LA 70504-1010 06532 Ankara USA Turkey e-mail: [email protected] e-mail: [email protected] 2000 Mathematics Subject Classification 13, 16; 17W, 18E Library of Congress Control Number: 2010927812 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISBN 978-3-0346-0006-4 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2010 Springer Basel AG P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Cover Design: Alexander Faust, Basel, Switzerland Printed in Germany ISBN 978-3-0346-0006-4 e-ISBN 978-3-0346-0007-1 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Preface ................................................................... vii T. Albu A Seventy Years Jubilee: The Hopkins-Levitzki Theorem . 1 G.F. Birkenmeier, J.K. Park and S. Tariq Rizvi A TheoryofHullsforRingsandModules ............................ 27 A.S. C¸evik, C. Ozel¨ and E.G. Karpuz Rewriting as a Special Case of Noncommutative Gr¨obner Bases Theory for the Affine Weyl Group An ................................ 73 C.J. Chikunji A Classification of a Certain Class of CompletelyPrimaryFiniteRings .................................... 83 M. Cohen and S. Westreich Higman Ideals and Verlinde-type Formulas forHopfAlgebras .................................................... 91 H.E. Heatherly and R.P. Tucci RightWeaklyRegularRings:A Survey .............................. 115 F. Kasch Substructures of Hom . 125 D. Keskin T¨ut¨unc¨u and S.H. Mohamed WeakLiftingModuleswithSmallRadical ........................... 129 L. Mao and N. Ding FP-injectiveComplexes ............................................. 135 M.J.NikmehrandF.Shaveisi A GeneralizationofHomologicalDimensions ......................... 143 P.F. Smith Additive Rank Functions and Chain Conditions . 149 vi Contents L. VaˇsandC.Papachristou ANoteon(α, β)-higher Derivations and their Extensions to ModulesofQuotients ................................................ 165 R. Wiegand and S. Wiegand PrimeIdealsinNoetherianRings:A Survey ......................... 175 Participants .............................................................. 195 Preface This volume is a collection of 13 peer reviewed papers consisting of expository/sur- vey articles and research papers by 24 authors. Many of these papers were presented at the International Conference on Ring and Module Theory held at Hacettepe University in Ankara, Turkey, during August 18–22, 2008. The selected papers and articles examine wide ranging and cutting edge de- velopments in various areas of Algebra including Ring Theory, Module Theory, Hopf Algebras, and Commutative Algebra. The survey articles are by well-known experts in their respective areas and provide an overview which is useful for re- searchers in the area, as well as, for researchers looking for new or related fields to investigate. The research papers give a taste of current research. We feel the variety of topics will be of interest to both graduate students and researchers. We wish to thank the large number of conference participants from over 20 countries, the contributors to this volume, and the referees. Encouragement and support from Hacettepe University, The Scientific and Technological Research Council of Turkey (TUB¨ ITAK)˙ and Republic of Turkey Ministry of Culture and Tourism are greatly appreciated. We also appreciate Evrim Akalan, Sevil Barın, Canan Celep Y¨ucel, Esra Demiry¨urek, Ozlem¨ Erdo˘gan, Fatih Karabacak, Didem Kavalcı, Mine Polat, Tu˘g¸ce Sivrikaya, Ay¸se S¨onmez, Figen Takıl, Muharrem Yavuz, Filiz Yıldız and U˘gur Y¨ucel for their assistance and efficient arrangement of the facilities which greatly contributed to the success of the conference. Finally, we must thank Erkan Afacan of Gazi University for his excellent job of typing and uniformizing manuscripts. December, 2009 The Editors Toma Albu, Bucharest, Romania Gary F. Birkenmeier, Lafayette, USA Ali Erdo˘gan, Ankara, Turkey Adnan Tercan, Ankara, Turkey Ring and Module Theory Trends in Mathematics, 1–26 c 2010 Springer Basel AG A Seventy Years Jubilee: The Hopkins-Levitzki Theorem Toma Albu Dedicated to the memory of Mark L. Teply (1942–2006) Abstract. The aim of this expository paper is to discuss various aspects of the Hopkins-Levitzki Theorem (H-LT), including the Relative H-LT, the Absolute or Categorical H-LT, the Latticial H-LT, as well as the Krull dimension-like H-LT. Mathematics Subject Classification (2000). Primary 16-06, 16P20, 16P40; Sec- ondary 16P70, 16S90, 18E15, 18E40. Keywords. Hopkins-Levitzki Theorem, Noetherian module, Artinian module, hereditary torsion theory, τ-Noetherian module, τ-Artinian module, quotient category, localization, Grothendieck category, modular lattice, upper contin- uous lattice, Krull dimension, dual Krull dimension. 1. Introduction In this expository paper we present a survey of the work done in the last forty years on various extensions of the Classical Hopkins-Levitzki Theorem: Relative, Absolute or Categorical, Latticial, and Krull dimension-like. We shall also illustrate a general strategy which consists on putting a module- theoretical theorem in a latticial frame, in order to translate that theorem to Grothendieck categories and module categories equipped with hereditary torsion theories. The author gratefully acknowledges partial financial support from the Grant ID-PCE 1190/2008 awarded by the Consiliul Nat¸ional al Cercet˘arii S¸tiint¸ifice ˆın ˆInv˘at¸˘amˆantul Superior (CNCSIS), Romˆania. 2T.Albu The (Molien-)Wedderburn-Artin Theorem One can say that the Modern Ring Theory begun in 1908, when Joseph Henry Maclagan Wedderburn (1882–1948) proved his celebrated Classification Theorem for finitely dimensional semi-simple algebras over a field F (see [49]). Before that, in 1893, Theodor Molien or Fedor Eduardovich Molin (1861–1941) proved the theorem for F = C (see [36]). In 1921, Emmy Noether (1882–1935) considers in her famous paper [42], for the first time in the literature, the Ascending Chain Condition (ACC) I1 ⊆ I2 ⊆···⊆In ⊆··· for ideals in a commutative ring R. In 1927, Emil Artin (1898–1962) introduces in [17] the Descending Chain Condition (DCC) I1 ⊇ I2 ⊇···⊇In ⊇··· for left/right ideals of a ring and extends the Wedderburn Theorem to rings satis- fying both the DCC and ACC for left/right ideals, observing that both ACC and DCC are a good substitute for finite dimensionality of algebras over a field: The (Molien-)Wedderburn-Artin Theorem. Aring R is semi-simple if and only if R is isomorphic to a finite direct product of full matrix rings over skew- fields ×···× R Mn1 (D1) Mnk (Dk). Recall that by a semi-simple ring one understands a ring R which is left (or right) Artinian and has Jacobson radical or prime radical zero. Since 1927, the (Molien-)Wedderburn-Artin Theorem became a cornerstone of the Noncommuta- tive Ring Theory. In 1929, Emmy Noether
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