# Ideals and Reality: Projective Modules and Number of Generators of Ideals

Springer Science & Business Media, Nov 24, 2004 - Mathematics - 336 pages
Besides giving an introduction to Commutative Algebra - the theory of c- mutative rings - this book is devoted to the study of projective modules and the minimal number of generators of modules and ideals. The notion of a module over a ring R is a generalization of that of a vector space over a field k. The axioms are identical. But whereas every vector space possesses a basis, a module need not always have one. Modules possessing a basis are called free. So a finitely generated free R-module is of the form Rn for some n E IN, equipped with the usual operations. A module is called p- jective, iff it is a direct summand of a free one. Especially a finitely generated R-module P is projective iff there is an R-module Q with P @ Q S Rn for some n. Remarkably enough there do exist nonfree projective modules. Even there are nonfree P such that P @ Rm S Rn for some m and n. Modules P having the latter property are called stably free. On the other hand there are many rings, all of whose projective modules are free, e. g. local rings and principal ideal domains. (A commutative ring is called local iff it has exactly one maximal ideal. ) For two decades it was a challenging problem whether every projective module over the polynomial ring k[X1,. . .

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### Contents

 II 1 IV 6 V 15 VII 17 VIII 20 IX 21 X 23 XII 27
 LV 149 LVI 152 LVII 154 LVIII 156 LIX 158 LX 162 LXI 164 LXII 168

 XIII 30 XIV 38 XVI 42 XVII 43 XVIII 45 XIX 46 XX 49 XXII 52 XXIII 56 XXIV 59 XXV 69 XXVII 74 XXVIII 75 XXIX 76 XXX 80 XXXI 82 XXXII 83 XXXIV 84 XXXV 87 XXXVII 89 XXXVIII 91 XXXIX 98 XL 101 XLI 105 XLIII 108 XLIV 114 XLV 121 XLVI 126 XLVIII 128 XLIX 129 L 131 LI 135 LII 136 LIII 140 LIV 143
 LXIII 171 LXIV 175 LXVI 180 LXVII 185 LXVIII 187 LXIX 189 LXX 194 LXXI 196 LXXII 200 LXXIII 203 LXXIV 205 LXXV 209 LXXVI 213 LXXVIII 216 LXXIX 221 LXXX 226 LXXXI 227 LXXXII 228 LXXXIII 231 LXXXIV 236 LXXXV 238 LXXXVI 241 LXXXVII 243 LXXXVIII 245 LXXXIX 248 XCI 251 XCII 253 XCIII 257 XCIV 264 XCV 269 XCVI 271 XCVII 289 XCVIII 325 XCIX 333 Copyright