Algebra: Chapter 0 (cid:52)(cid:69)(cid:83)(cid:80)(cid:83)(cid:3)(cid:37)(cid:80)(cid:89)(cid:74)(cid:189) Graduate Studies in Mathematics Volume 104 American Mathematical Society Algebra: Chapter 0 Algebra: Chapter 0 Paolo Aluffi Graduate Studies in Mathematics Volume 104 American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2010 Mathematics Subject Classification. Primary00–01;Secondary 12–01,13–01,15–01, 18–01, 20–01. For additional informationand updates on this book, visit www.ams.org/bookpages/gsm-104 Library of Congress Cataloging-in-Publication Data Aluffi,Paolo,1960– Algebra: chapter0/PaoloAluffi. p.cm. —(Graduatestudiesinmathematics;v.104) Includesindex. ISBN978-0-8218-4781-7(alk.paper) 1.Algebra—Textbooks. I.Title. QA154.3.A527 2009 512—dc22 2009004043 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for useinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationin reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink(cid:2) service. Formoreinformation,pleasevisit: http://www.ams.org/rightslink. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases, requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s). Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe firstpageofeacharticlewithinproceedingsvolumes. (cid:2)c 2009bytheAmericanMathematicalSociety. Allrightsreserved. ReprintedwithcorrectionsbytheAmericanMathematicalSociety,2016. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 1098765432 212019181716 Contents Preface to the second printing xv Introduction xvii Chapter I. Preliminaries: Set theory and categories 1 §1. Naive set theory 1 1.1. Sets 1 1.2. Inclusion of sets 3 1.3. Operations between sets 4 1.4. Disjoint unions, products 5 1.5. Equivalence relations, partitions, quotients 6 Exercises 8 §2. Functions between sets 8 2.1. Definition 8 2.2. Examples: Multisets, indexed sets 10 2.3. Composition of functions 10 2.4. Injections, surjections, bijections 11 2.5. Injections, surjections, bijections: Second viewpoint 12 2.6. Monomorphisms and epimorphisms 14 2.7. Basic examples 15 2.8. Canonical decomposition 15 2.9. Clarification 16 Exercises 17 §3. Categories 18 3.1. Definition 18 3.2. Examples 20 Exercises 26 v vi Contents §4. Morphisms 27 4.1. Isomorphisms 27 4.2. Monomorphisms and epimorphisms 29 Exercises 30 §5. Universal properties 31 5.1. Initial and final objects 31 5.2. Universal properties 33 5.3. Quotients 33 5.4. Products 35 5.5. Coproducts 36 Exercises 38 Chapter II. Groups, first encounter 41 §1. Definition of group 41 1.1. Groups and groupoids 41 1.2. Definition 42 1.3. Basic properties 43 1.4. Cancellation 45 1.5. Commutative groups 45 1.6. Order 46 Exercises 48 §2. Examples of groups 49 2.1. Symmetric groups 49 2.2. Dihedral groups 52 2.3. Cyclic groups and modular arithmetic 54 Exercises 56 §3. The category Grp 58 3.1. Group homomorphisms 58 3.2. Grp: Definition 59 3.3. Pause for reflection 60 3.4. Products et al. 61 3.5. Abelian groups 62 Exercises 63 §4. Group homomorphisms 64 4.1. Examples 64 4.2. Homomorphisms and order 66 4.3. Isomorphisms 66 4.4. Homomorphisms of abelian groups 68 Exercises 69 §5. Free groups 70 5.1. Motivation 70 5.2. Universal property 71 5.3. Concrete construction 72 5.4. Free abelian groups 75 Contents vii Exercises 78 §6. Subgroups 79 6.1. Definition 79 6.2. Examples: Kernel and image 80 6.3. Example: Subgroup generated by a subset 81 6.4. Example: Subgroups of cyclic groups 82 6.5. Monomorphisms 84 Exercises 85 §7. Quotient groups 88 7.1. Normal subgroups 88 7.2. Quotient group 89 7.3. Cosets 90 7.4. Quotient by normal subgroups 92 7.5. Example 94 7.6. kernel ⇐⇒ normal 95 Exercises 95 §8. Canonical decomposition and Lagrange’s theorem 96 8.1. Canonical decomposition 97 8.2. Presentations 99 8.3. Subgroups of quotients 100 8.4. HK/H vs. K/(H ∩K) 101 8.5. The index and Lagrange’s theorem 102 8.6. Epimorphisms and cokernels 104 Exercises 105 §9. Group actions 108 9.1. Actions 108 9.2. Actions on sets 109 9.3. Transitive actions and the category G-Set 110 Exercises 113 §10. Group objects in categories 115 10.1. Categorical viewpoint 115 Exercises 117 Chapter III. Rings and modules 119 §1. Definition of ring 119 1.1. Definition 119 1.2. First examples and special classes of rings 121 1.3. Polynomial rings 124 1.4. Monoid rings 126 Exercises 127 §2. The category Ring 129 2.1. Ring homomorphisms 129 2.2. Universal property of polynomial rings 130 2.3. Monomorphisms and epimorphisms 132 viii Contents 2.4. Products 133 2.5. End (G) 134 Ab Exercises 136 §3. Ideals and quotient rings 138 3.1. Ideals 138 3.2. Quotients 139 3.3. Canonical decomposition and consequences 141 Exercises 143 §4. Ideals and quotients: Remarks and examples. Prime and maximal ideals 144 4.1. Basic operations 144 4.2. Quotients of polynomial rings 146 4.3. Prime and maximal ideals 150 Exercises 153 §5. Modules over a ring 156 5.1. Definition of (left-)R-module 156 5.2. The category R-Mod 158 5.3. Submodules and quotients 160 5.4. Canonical decomposition and isomorphism theorems 162 Exercises 163 §6. Products, coproducts, etc., in R-Mod 164 6.1. Products and coproducts 164 6.2. Kernels and cokernels 166 6.3. Free modules and free algebras 167 6.4. Submodule generated by a subset; Noetherian modules 169 6.5. Finitely generated vs. finite type 171 Exercises 172 §7. Complexes and homology 174 7.1. Complexes and exact sequences 174 7.2. Split exact sequences 177 7.3. Homology and the snake lemma 178 Exercises 183 Chapter IV. Groups, second encounter 187 §1. The conjugation action 187 1.1. Actions of groups on sets, reminder 187 1.2. Center, centralizer, conjugacy classes 189 1.3. The Class Formula 190 1.4. Conjugation of subsets and subgroups 191 Exercises 193 §2. The Sylow theorems 194 2.1. Cauchy’s theorem 194 2.2. Sylow I 196 2.3. Sylow II 197 Contents ix 2.4. Sylow III 199 2.5. Applications 200 Exercises 202 §3. Composition series and solvability 205 3.1. The Jordan-Ho¨lder theorem 205 3.2. Composition factors; Schreier’s theorem 207 3.3. The commutator subgroup, derived series, and solvability 210 Exercises 213 §4. The symmetric group 214 4.1. Cycle notation 214 4.2. Type and conjugacy classes in S 216 n 4.3. Transpositions, parity, and the alternating group 219 4.4. Conjugacy in A ; simplicity of A and solvability of S 220 n n n Exercises 224 §5. Products of groups 226 5.1. The direct product 226 5.2. Exact sequences of groups; extension problem 228 5.3. Internal/semidirect products 230 Exercises 233 §6. Finite abelian groups 234 6.1. Classification of finite abelian groups 234 6.2. Invariant factors and elementary divisors 237 6.3. Application: Finite subgroups of multiplicative groups of fields 239 Exercises 240 Chapter V. Irreducibility and factorization in integral domains 243 §1. Chain conditions and existence of factorizations 244 1.1. Noetherian rings revisited 244 1.2. Prime and irreducible elements 246 1.3. Factorization into irreducibles; domains with factorizations 248 Exercises 249 §2. UFDs, PIDs, Euclidean domains 251 2.1. Irreducible factors and greatest common divisor 251 2.2. Characterization of UFDs 253 2.3. PID =⇒ UFD 254 2.4. Euclidean domain =⇒ PID 255 Exercises 258 §3. Intermezzo: Zorn’s lemma 261 3.1. Set theory, reprise 261 3.2. Application: Existence of maximal ideals 264 Exercises 265 §4. Unique factorization in polynomial rings 267 4.1. Primitivity and content; Gauss’s lemma 268