Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules
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Modern Birkhauser¨ Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkhauser¨ in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain accessible to new generations of students, scholars, and researchers. Alberto Facchini Module Theory Endomorphism rings and direct sum decompositions in some classes of modules Reprint of the 1998 Edition Alberto Facchini Department of Mathematics University of Padova Padova Italy ISBN 978-3-0348-0302 - 1 e-ISBN 978-3-0348-0303 - 8 DOI 10.1007/978-3-0348-0303-8 Springer Basel Dordrecht Heidelberg London New York Library of Congress Control Number: 2012930380 2010 Mathematics Subject Classification: 16D70, 16-02, 16S50, 16P20, 16P60, 16P70 © Springer Basel AG 1998 Reprint of the 1st edition 1998 by Birkhäuser Verlag, Switzerland Originally published as volume 167 in the Progress in Mathematics series 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. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com) Contents Preface .................................... ix List of Symbols ................................ xiii 1BasicConcepts 1.1Semisimpleringsandmodules................... 1 1.2Localandsemilocalrings..................... 5 1.3Serialringsandmodules...................... 9 1.4Pureexactsequences........................ 12 1.5 Finitely definable subgroups and pure-injective modules . 18 1.6 The category (RFP,Ab)...................... 25 1.7Σ-pure-injectivemodules...................... 28 1.8NotesonChapter1......................... 31 2 The Krull-Schmidt-Remak-Azumaya Theorem 2.1Theexchangeproperty....................... 33 2.2 Indecomposable modules with the exchange property . 36 2.3 Isomorphic refinements of finite direct sumdecompositions........................ 39 2.4 The Krull-Schmidt-Remak-Azumaya Theorem . 43 2.5Applications............................. 45 2.6Goldiedimensionofamodularlattice.............. 51 2.7Goldiedimensionofamodule................... 55 2.8DualGoldiedimensionofamodule................ 57 2.9 ℵ-small modules and ℵ-closedclasses............... 61 2.10 Direct sums of ℵ-smallmodules.................. 65 2.11TheLoewyseries.......................... 68 2.12 Artinian right modules over commutative or rightnoetherianrings....................... 70 2.13NotesonChapter2......................... 71 v vi Contents 3 Semiperfect Rings 3.1Projectivecoversandliftingidempotents............. 75 3.2Semiperfectrings.......................... 79 3.3Modulesoversemiperfectrings.................. 82 3.4 Finitely presented and Fitting modules . 86 3.5 Finitely presented modules over serial rings . 92 3.6NotesonChapter3......................... 96 4 Semilocal Rings 4.1TheCamps-DicksTheorem.................... 99 4.2 Modules with semilocal endomorphism ring . 103 4.3Examples.............................. 108 4.4NotesonChapter4......................... 110 5 Serial Rings 5.1Chainringsandrightchainrings................. 113 5.2Modulesoverartinianserialrings................. 115 5.3 Nonsingular and semihereditary serial rings . 118 5.4Noetherianserialrings....................... 121 5.5NotesonChapter5......................... 128 6QuotientRings 6.1Quotientringsofarbitraryrings................. 129 6.2NilsubringsofrightGoldierings................. 132 6.3Reducedrank............................ 134 6.4Localizationinchainrings..................... 140 6.5Localizablesystemsinaserialring................ 142 6.6Anexample............................. 147 6.7Primeidealsinserialrings..................... 149 6.8Goldiesemiprimeideals...................... 152 6.9 Diagonalization of matrices . .................. 155 6.10Oresetsinserialrings....................... 158 6.11 Goldie semiprime ideals and maximal Ore sets . 161 6.12Classicalquotientringofaserialring.............. 162 6.13NotesonChapter6......................... 169 7 Krull Dimension and Serial Rings 7.1Deviationofaposet........................ 171 7.2 Krull dimension of arbitrary modules and rings . 175 7.3 Nil subrings of rings with right Krull dimension . 177 7.4 Transfinite powers of the Jacobson radical . 182 7.5 Structure of serial rings of finite Krull dimension . 187 7.6NotesonChapter7......................... 191 Contents vii 8 Krull-Schmidt Fails for Finitely Generated Modules and Artinian Modules 8.1 Krull-Schmidt fails for finitely generated modules . 193 8.2Krull-Schmidtfailsforartinianmodules............. 201 8.3NotesonChapter8......................... 205 9 Biuniform Modules 9.1Firstpropertiesofbiuniformmodules.............. 209 9.2Sometechnicallemmas....................... 212 9.3Asufficientcondition........................ 216 9.4 Weak Krull-Schmidt Theorem for biuniform modules . 218 9.5 Krull-Schmidt holds for finitely presented modules overchainrings........................... 220 9.6 Krull-Schmidt fails for finitely presented modules overserialrings........................... 222 9.7 Further examples of biuniform modules of type 1 . 225 9.8Quasi-smalluniserialmodules................... 227 9.9 A necessary condition for families of uniserial modules . 231 9.10NotesonChapter9......................... 233 10 Σ-pure-injective Modules and Artinian Modules 10.1 Rings with a faithful Σ-pure-injective module . 237 10.2 Rings isomorphic to endomorphism rings of artinian modules . 240 10.3Distributivemodules........................ 244 10.4Σ-pure-injectivemodulesoverchainrings............ 249 10.5HomogeneousΣ-pure-injectivemodules............. 251 10.6 Krull dimension and Σ-pure-injective modules . 254 10.7 Serial rings that are endomorphism rings ofartinianmodules......................... 257 10.8 Localizable systems and Σ-pure-injective modules overserialrings........................... 260 10.9NotesonChapter10........................ 264 11 Open Problems .............................. 267 Bibliography ................................. 271 Index ..................................... 283 Preface This expository monograph was written for three reasons. Firstly, we wanted to present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the “Krull-Schmidt Theorem” holds for ar- tinian modules. The problem remained open for 63 years: its solution, a negative answer to Krull’s question, was published only in 1995 (see [Facchini, Herbera, Levy and V´amos]). Secondly, we wanted to present the answer to a question posed by Warfield in 1975 [Warfield 75]. He proved that every finitely pre- sented module over a serial ring is a direct sum of uniserial modules, and asked if such a decomposition was unique. In other words, Warfield asked whether the “Krull-Schmidt Theorem” holds for serial modules. The solution to this problem, a negative answer again, appeared in [Facchini 96]. Thirdly, the so- lution to Warfield’s problem shows interesting behavior, a rare phenomenon in the history of Krull-Schmidt type theorems. Essentially, the Krull-Schmidt Theorem holds for some classes of modules and not for others. When it does hold, any two indecomposable decompositions are uniquely determined up to a permutation, and when it does not hold for a class of modules, this is proved via an example. For serial modules the Krull-Schmidt Theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations. We wanted to present such a phenomenon to a wider math- ematical audience. Apart from these three reasons, we present in this book various topics of module theory and ring theory, some of which are now considered classical (like Goldie dimension, semiperfect rings, Krull dimension, rings of quotients, and their applications) whereas others are more specialized (like dual Goldie dimension, semilocal endomorphism rings, serial rings and modules, exchange property, Σ-pure-injective modules). We now consider the three reasons above in more detail. 1) Krull’s problem. The classical Krull-Schmidt Theorem says that if MR is a right module of finite composition length over a ring R and MR = A1 ⊕···⊕An = B1 ⊕···⊕Bm are two decompositions of MR as direct sums of indecomposable modules, then ix x Preface ∼ n = m and, after a suitable renumbering of the summands, Ai = Bi for every i =1,...,n. In the paper [Krull 32, pp. 37–38] Krull recalls this theorem and asks whether the result remains true for artinian modules; that is, whether A1 ⊕···⊕An = B1 ⊕···⊕Bm with each Ai and Bj an indecomposable artinian right module over a ring R, implies that m = n and, after a renumbering of ∼ the summands, Ai = Bi for each i. During the years Krull’s question was not forgotten (see for instance [Levy, p. 660]), and various partial results were proved. For instance, [Warfield 69a, Proposition 5] showed that the answer is “yes” when the ring R is either right noetherian or commutative (Proposition 2.63). He did this by showing that, over any ring, every artinian indecomposable module with Loewy length ≤ ω has a local endomorphism ring. By the Krull-Schmidt-Remak-Azumaya