Rings of Quotients: An Introduction to Methods of Ring Theory
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1975
|
| Schriftenreihe: | Die Grundlehren der mathematischen Wissenschaften, Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete
217 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | 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) |
| Beschreibung: | 1 Online-Ressource (309p) |
| ISBN: | 9783642660665 9783642660689 |
| ISSN: | 0072-7830 |
| DOI: | 10.1007/978-3-642-66066-5 |
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| 500 | |a 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) | ||
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| discipline | Mathematik |
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| isbn | 9783642660665 9783642660689 |
| issn | 0072-7830 |
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| spelling | Stenström, Bo Verfasser aut Rings of Quotients An Introduction to Methods of Ring Theory by Bo Stenström Berlin, Heidelberg Springer Berlin Heidelberg 1975 1 Online-Ressource (309p) txt rdacontent c rdamedia cr rdacarrier Die Grundlehren der mathematischen Wissenschaften, Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete 217 0072-7830 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) Mathematics Mathematics, general Mathematik Ring Mathematik (DE-588)4128084-2 gnd rswk-swf Quotientenring (DE-588)4176745-7 gnd rswk-swf Quotientenring (DE-588)4176745-7 s 1\p DE-604 Ring Mathematik (DE-588)4128084-2 s 2\p DE-604 https://doi.org/10.1007/978-3-642-66066-5 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Stenström, Bo Rings of Quotients An Introduction to Methods of Ring Theory Mathematics Mathematics, general Mathematik Ring Mathematik (DE-588)4128084-2 gnd Quotientenring (DE-588)4176745-7 gnd |
| subject_GND | (DE-588)4128084-2 (DE-588)4176745-7 |
| title | Rings of Quotients An Introduction to Methods of Ring Theory |
| title_auth | Rings of Quotients An Introduction to Methods of Ring Theory |
| title_exact_search | Rings of Quotients An Introduction to Methods of Ring Theory |
| title_full | Rings of Quotients An Introduction to Methods of Ring Theory by Bo Stenström |
| title_fullStr | Rings of Quotients An Introduction to Methods of Ring Theory by Bo Stenström |
| title_full_unstemmed | Rings of Quotients An Introduction to Methods of Ring Theory by Bo Stenström |
| title_short | Rings of Quotients |
| title_sort | rings of quotients an introduction to methods of ring theory |
| title_sub | An Introduction to Methods of Ring Theory |
| topic | Mathematics Mathematics, general Mathematik Ring Mathematik (DE-588)4128084-2 gnd Quotientenring (DE-588)4176745-7 gnd |
| topic_facet | Mathematics Mathematics, general Mathematik Ring Mathematik Quotientenring |
| url | https://doi.org/10.1007/978-3-642-66066-5 |
| work_keys_str_mv | AT stenstrombo ringsofquotientsanintroductiontomethodsofringtheory |