# Free Ideal Rings and Localization in General Rings

Cambridge University Press, Jun 8, 2006 - Mathematics
Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization which is treated for general rings but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends in a historical note.

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### Contents

 I 1 II 7 III 12 IV 19 V 25 VI 33 VII 37 VIII 47
 XLIII 251 XLIV 261 XLV 263 XLVI 264 XLVII 269 XLVIII 272 XLIX 281 L 290

 IX 52 X 58 XI 60 XII 66 XIII 73 XIV 77 XV 86 XVI 98 XVII 105 XVIII 107 XX 110 XXI 113 XXII 124 XXIII 131 XXIV 141 XXV 145 XXVI 153 XXVII 156 XXVIII 171 XXIX 176 XXX 183 XXXI 186 XXXIII 192 XXXIV 199 XXXV 207 XXXVI 214 XXXVII 223 XXXVIII 225 XXXIX 231 XL 237 XLI 243 XLII 247
 LI 300 LII 304 LIII 311 LIV 320 LV 326 LVI 329 LVII 331 LVIII 340 LIX 351 LX 355 LXI 357 LXII 367 LXIII 374 LXIV 379 LXV 387 LXVI 396 LXVII 407 LXVIII 410 LXIX 411 LXX 418 LXXI 428 LXXII 437 LXXIII 444 LXXIV 455 LXXV 466 LXXVI 482 LXXVII 491 LXXVIII 500 LXXIX 511 LXXX 515 Copyright

### Popular passages

Page 531 - Since a module is projective if and only if it is a direct summand of a free module, we have: Corollary 1.6.5.
Page 542 - Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups. Trans. Amer. Math. Soc. 351 (1999), 1531-1550.
Page 542 - Bergman, Hereditary commutative rings and centres of hereditary rings, Proc. London Math. Soc. 23 (1971), 214-236. 4. J. Dauns and K.. Hofmann, The representation of biregular rings by sheaves, Math.
Page 521 - ... if we can pass from one to the other by a series of perspectivities.
Page 541 - Infinite primes and unique factorization in a principal right ideal domain. Trans. Amer. Math. Soc. 141 (1969), 245-254.
Page 522 - L is a distributive lattice with 0 and 1 in which every element has a complement, we can regard the process of associating with each element x its complement x' as a unary operator, ie an operator with one argument.
Page 548 - A free algebra can be free as a module over a non-free subalgebra. Bull.