Author: B. Stenström
Publisher: Springer Science & Business Media
ISBN: 3642660665
Category : Mathematics
Languages : en
Pages : 319
Book Description
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).
Rings of Quotients
Author: B. Stenström
Publisher: Springer Science & Business Media
ISBN: 3642660665
Category : Mathematics
Languages : en
Pages : 319
Book Description
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).
Publisher: Springer Science & Business Media
ISBN: 3642660665
Category : Mathematics
Languages : en
Pages : 319
Book Description
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).
Injective Modules and Injective Quotient Rings
Author: Carl Faith
Publisher: CRC Press
ISBN: 1000657310
Category : Mathematics
Languages : en
Pages : 120
Book Description
First published in 1982. These lectures are in two parts. Part I, entitled injective Modules Over Levitzki Rings, studies an injective module E and chain conditions on the set A^(E,R) of right ideals annihilated by subsets of E. Part II is on the subject of (F)PF, or (finitely) pseudo-Frobenius, rings [i.e., all (finitely generated) faithful modules generate the category mod-R of all R-modules]. (The PF rings had been introduced by Azumaya as a generalization of quasi-Frobenius rings, but FPF includes infinite products of Prufer domains, e.g., Z w .)
Publisher: CRC Press
ISBN: 1000657310
Category : Mathematics
Languages : en
Pages : 120
Book Description
First published in 1982. These lectures are in two parts. Part I, entitled injective Modules Over Levitzki Rings, studies an injective module E and chain conditions on the set A^(E,R) of right ideals annihilated by subsets of E. Part II is on the subject of (F)PF, or (finitely) pseudo-Frobenius, rings [i.e., all (finitely generated) faithful modules generate the category mod-R of all R-modules]. (The PF rings had been introduced by Azumaya as a generalization of quasi-Frobenius rings, but FPF includes infinite products of Prufer domains, e.g., Z w .)
Lectures on Injective Modules and Quotient Rings
Author: Carl Faith
Publisher: Springer
ISBN: 3540355510
Category : Mathematics
Languages : en
Pages : 158
Book Description
Publisher: Springer
ISBN: 3540355510
Category : Mathematics
Languages : en
Pages : 158
Book Description
Rings and Modules of Quotients
Author: B. Stenström
Publisher: Springer
ISBN: 3540370021
Category : Mathematics
Languages : en
Pages : 143
Book Description
Publisher: Springer
ISBN: 3540370021
Category : Mathematics
Languages : en
Pages : 143
Book Description
Injective Modules and Injective Quotient Rings
Author: Carl Faith
Publisher: CRC Press
ISBN: 1000673030
Category : Mathematics
Languages : en
Pages : 124
Book Description
First published in 1982. These lectures are in two parts. Part I, entitled injective Modules Over Levitzki Rings, studies an injective module E and chain conditions on the set A^(E,R) of right ideals annihilated by subsets of E. Part II is on the subject of (F)PF, or (finitely) pseudo-Frobenius, rings [i.e., all (finitely generated) faithful modules generate the category mod-R of all R-modules]. (The PF rings had been introduced by Azumaya as a generalization of quasi-Frobenius rings, but FPF includes infinite products of Prufer domains, e.g., Z w .)
Publisher: CRC Press
ISBN: 1000673030
Category : Mathematics
Languages : en
Pages : 124
Book Description
First published in 1982. These lectures are in two parts. Part I, entitled injective Modules Over Levitzki Rings, studies an injective module E and chain conditions on the set A^(E,R) of right ideals annihilated by subsets of E. Part II is on the subject of (F)PF, or (finitely) pseudo-Frobenius, rings [i.e., all (finitely generated) faithful modules generate the category mod-R of all R-modules]. (The PF rings had been introduced by Azumaya as a generalization of quasi-Frobenius rings, but FPF includes infinite products of Prufer domains, e.g., Z w .)
Rings of Quotients and Quotient Modules
Author: Cathleen Clare Real
Publisher:
ISBN:
Category : Rings (Algebra)
Languages : en
Pages : 210
Book Description
Publisher:
ISBN:
Category : Rings (Algebra)
Languages : en
Pages : 210
Book Description
Quotient Rings of Non-commutative Rings
Author: Sie-hua Chen
Publisher:
ISBN:
Category : Rings (Algebra)
Languages : en
Pages : 31
Book Description
Publisher:
ISBN:
Category : Rings (Algebra)
Languages : en
Pages : 31
Book Description
Quotient Rings and Embeddings of Noetherian Rings
Quasi-Frobenius Rings and Generalizations
Author: H. Tachikawa
Publisher: Springer
ISBN: 354037812X
Category : Mathematics
Languages : en
Pages : 184
Book Description
As each animal unbuttons its buttons another animal appears.
Publisher: Springer
ISBN: 354037812X
Category : Mathematics
Languages : en
Pages : 184
Book Description
As each animal unbuttons its buttons another animal appears.
Rings and Modules of Quotients
Author: B. Stenstrom
Publisher:
ISBN: 9783662195826
Category :
Languages : en
Pages : 148
Book Description
Publisher:
ISBN: 9783662195826
Category :
Languages : en
Pages : 148
Book Description