# Rings of Quotients

## An Introduction to Methods of Ring Theory

• Bo Stenström
Book

Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 217)

1. Front Matter
Pages I-VIII
2. Bo Stenström
Pages 1-3
3. Bo Stenström
Pages 4-4
4. Bo Stenström
Pages 5-49
5. Bo Stenström
Pages 50-62
6. Bo Stenström
Pages 63-81
7. Bo Stenström
Pages 82-113
8. Bo Stenström
Pages 114-135
9. Bo Stenström
Pages 136-159
10. Bo Stenström
Pages 160-178
11. Bo Stenström
Pages 179-194
12. Bo Stenström
Pages 195-212
13. Bo Stenström
Pages 213-224
14. Bo Stenström
Pages 225-243
15. Bo Stenström
Pages 244-261
16. Bo Stenström
Pages 262-272
17. Bo Stenström
Pages 273-282
18. Bo Stenström
Pages 283-294
19. Back Matter
Pages 295-312

### Introduction

The theory of rings of quotients has its origin in the work of (j). Ore and K. Asano on the construction of the total ring of fractions, in the 1930's and 40's. But the subject did not really develop until the end of the 1950's, when a number of important papers appeared (by R. E. Johnson, Y. Utumi, A. W. Goldie, P. Gabriel, J. Lambek, and others). Since then the progress has been rapid, and the subject has by now attained a stage of maturity, where it is possible to make a systematic account of it (which is the purpose of this book). The most immediate example of a ring of quotients is the field of fractions Q of a commutative integral domain A. It may be characterized by the two properties: (i) For every qEQ there exists a non-zero SEA such that qSEA. (ii) Q is the maximal over-ring of A satisfying condition (i). The well-known construction of Q can be immediately extended to the case when A is an arbitrary commutative ring and S is a multiplicatively closed set of non-zero-divisors of A. In that case one defines the ring of fractions Q = A [S-l] as consisting of pairs (a, s) with aEA and SES, with the declaration that (a, s)=(b, t) if there exists UES such that uta = usb. The resulting ring Q satisfies (i), with the extra requirement that SES, and (ii).

### Keywords

Adjoint functor Coproduct Prime Quotientenring Rings algebra colimit

#### Authors and affiliations

• Bo Stenström
• 1
1. 1.Matematiska InstitutionenStockholms UniversitetSweden

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-66066-5
• Copyright Information Springer-Verlag Berlin Heidelberg 1975
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-66068-9
• Online ISBN 978-3-642-66066-5
• Series Print ISSN 0072-7830
• Buy this book on publisher's site