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 ğ Ali Erdo an 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 Theory of Hulls for Rings and Modules ............................ 27 A.S. C¸evik, C. O¨zel and E.G. Karpuz Rewriting as a Special Case of Noncommutative Gr¨obner Bases (cid:2) Theory for the Affine Weyl Group A ................................ 73 n C.J. Chikunji A Classification of a Certain Class of Completely Primary Finite Rings .................................... 83 M. Cohen and S. Westreich Higman Ideals and Verlinde-type Formulas for Hopf Algebras .................................................... 91 H.E. Heatherly and R.P. Tucci Right Weakly Regular Rings: A Survey .............................. 115 F. Kasch Substructures of Hom ............................................... 125 D. Keskin Tu¨tu¨ncu¨ and S.H. Mohamed Weak Lifting Modules with Small Radical ........................... 129 L. Mao and N. Ding FP-injective Complexes ............................................. 135 M.J. Nikmehr and F. Shaveisi A Generalization of Homological Dimensions ......................... 143 P.F. Smith Additive Rank Functions and Chain Conditions ...................... 149 vi Contents L. Vaˇs and C. Papachristou A Note on (α,β)-higher Derivations and their Extensions to Modules of Quotients ................................................ 165 R. Wiegand and S. Wiegand Prime Ideals in Noetherian Rings: A Survey ......................... 175 Participants .............................................................. 195 Preface Thisvolumeisacollectionof13peerreviewedpapersconsistingofexpository/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 andsupportfromHacettepeUniversity,TheScientificandTechnologicalResearch Council of Turkey (TU¨BI˙TAK) and Republic of Turkey Ministry of Culture and Tourism are greatly appreciated. We also appreciate Evrim Akalan, Sevil Barın, Canan Celep Yu¨cel, Esra Demiryu¨rek, O¨zlem Erdog˘an, Fatih Karabacak, Didem Kavalcı,MinePolat,Tug˘¸ceSivrikaya,Ay¸seS¨onmez,FigenTakıl,MuharremYavuz, Filiz Yıldız and Ug˘ur Yu¨cel for their assistance and efficient arrangement of the facilities which greatly contributed to the success of the conference. Finally, we mustthank ErkanAfacan ofGaziUniversity for his excellentjob of typing and uniformizing manuscripts. December, 2009 The Editors Toma Albu, Bucharest, Romania Gary F. Birkenmeier, Lafayette, USA Ali Erdog˘an,Ankara, Turkey Adnan Tercan, Ankara, Turkey RingandModuleTheory TrendsinMathematics,1–26 (cid:2)c 2010SpringerBaselAG A Seventy Years Jubilee: The Hopkins-Levitzki Theorem Toma Albu Dedicated tothe memory of Mark L. Teply(1942–2006) Abstract. Theaimofthisexpositorypaperistodiscussvariousaspectsofthe Hopkins-LevitzkiTheorem(H-LT),includingtheRelativeH-LT,theAbsolute or Categorical H-LT, the Latticial H-LT, as well as the Krull dimension-like H-LT. MathematicsSubjectClassification(2000).Primary16-06,16P20,16P40;Sec- ondary 16P70, 16S90, 18E15, 18E40. Keywords. Hopkins-LevitzkiTheorem, Noetherian module, Artinian module, hereditarytorsiontheory, τ-Noetherianmodule, τ-Artinianmodule,quotient category, localization, Grothendieck category, modular lattice, upper contin- uous lattice, Krulldimension, dualKrull 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. Weshallalsoillustrateageneralstrategy whichconsistsonputtingamodule- theoretical theorem in a latticial frame, in order to translate that theorem to Grothendieck categories and module categories equipped with hereditary torsion theories. TheauthorgratefullyacknowledgespartialfinancialsupportfromtheGrantID-PCE1190/2008 awarded bytheConsiliulNa¸tional alCercet˘arii S¸tiin¸tificeˆınˆInv˘a¸t˘amˆantul Superior (CNCSIS), Romaˆnia. 2 T. 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 dimensionalsemi-simple algebras overa 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/rightideals ofa ring andextends the WedderburnTheoremto 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-simpleifand only if R is isomorphic to a finite direct product of full matrix rings over skew- fields R(cid:4)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 observes (see [43, p. 643]) that the ACC in Artin’s extension of the Wedderburn Theorem can be omitted: Im II. Kapitel werden die Wedderburnschen Resultate neu gewonnen und weitergefu¨rt, .... Und zwar zeigt es sich das der “Vielfachenkettensatz” fu¨r Rechtsideale oder die damit identische “Minimalbedingung” (in jeder Menge von Rechtsidealen gibt es mindestens ein – in der Menge – minimales) als Endlichkeitsbedingung ausreicht (Die Wedderburn- schen Schlußweissen lassen sich u¨bertragen wenn “Doppelkettensatz” vorausgesezt wird. Vgl. E. Artin [17]). It took, however, ten years until it has been proved that always the DCC in a unital ring implies the ACC. The Classical Hopkins-Levitzki Theorem (H-LT) One of the most lovely result in Ring Theory is the Hopkins-Levitzki Theorem, abbreviatedH-LT.Thistheorem,sayingthatanyrightArtinianringwithidentity is right Noetherian, has been proved independently in 1939 by Charles Hopkins The Hopkins-Levitzki Theorem 3 [27]1 (1902–1939)for left ideals and by Jacob Levitzki [31]2 (1904–1956)for right ideals. Almost surely, the fact that the DCC implies the ACC for one-sidedideals in a unital ring was unknown to both E. Noether and E. Artin when they wrote their pioneering papers on chain conditions in the 1920’s. An equivalent form of the H-LT, referred in the sequel also as the Classical H-LT, is the following one: Classical H-LT. Let R be a right Artinian ring with identity, and let M be a R right module. Then M is an Artinian module if and only if M is a Noetherian R R module. Proof. The standard proof of this theorem, as well as the original one of Hopkins [27, Theorem 6.4] for M = R, uses the Jacobson radical J of R. SinceR is right Artinian,J isnilpotentandthequotientringR/J isasemi-simplering.Letnbea positiveintegersuchthatJn =0,andconsiderthedescendingchainofsubmodules of M R M ⊇MJ ⊇MJ2 ⊇···⊇MJn−1 ⊇MJn =0. Since the quotients MJk/MJk+1 are killed by J, k = 0, 1, ... ,n − 1, each MJk/MJk+1 becomes a right module over the semi-simple ring R/J, so each MJk/MJk+1 is a semi-simple (R/J)-module. Now,observethatM isArtinian(resp.Noetherian) ⇐⇒ all MJk/MJk+1 R are Artinian (resp. Noetherian) R (or R/J)-modules. Since a semi-simple module is Artinian if and only if it is Noetherian, it follows that M is Artinian if and R only if it is Noetherian, which finishes the proof. (cid:2) Extensions of the H-LT In the last fifty years, especially in the 1970’s, 1980’s, and 1990’s the (Classical) H-LT has been generalized and dualized as follows: 1957 Fuchs [21] shows that a left Artinian ring A, not necessarily unital, is Noetherian if and only if the additive group of A contains no subgroup isomorphic to the Pru¨fer quasi-cyclic p-group Zp∞. 1Infact,heprovedthatanyleftArtinianring(calledbyhimMLI ring)withleftorrightidentity isleftNoetherian(seeHopkins[27,Theorems6.4and6.7]). 2Theresultishowever,surprisingly,neitherstatednorprovedinhispaper,thoughinthelitera- ture,includingourpapers,theHopkins’TheoremisalsowronglyattributedtoLevitzki.Actually, whatLevitzkiprovedwasthattheACCissuperfluousinmostofthemainresultsoftheoriginal paper of Artin [17] assuming both the ACC and DCC for right ideals of a ring. This is also very clearly stated in the Introduction of his paper: “In the present note it is shown that the maximum condition can be omitted without affecting the results achieved by Artin.” Note that Levitzki considers rings which are not necessarily unital, so anyway it seems that he was even not awareabout DCC impliesACC inunital rings;this implicationdoes not holdingeneral in nonunitalrings,astheexampleoftheringwithzeromultiplicationassociatedwithanyPr¨ufer quasi-cyclic p-group Zp∞ shows. Note also that though all sources in the literature, including MathematicalReviews,indicate1939astheyearofappearanceofLevitzki’spaperinCompositia Mathematica,thefreereprintofthepaperavailableathttp://www.numdam.org indicates1940as theyearwhenthepaperhasbeenpublished.