Free Ideal Rings and Localization in General Rings

Front Cover
Cambridge University Press, Jun 8, 2006 - Mathematics
0 Reviews
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.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

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

Other editions - View all

Common terms and phrases

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.

About the author (2006)

Paul Cohn is a Emeritus Professor of Mathematics at the University of London and Honorary Research Fellow at University College London.

Bibliographic information