This book gathers, in a beautifully structured way, recent findings on chain conditions in commutative algebra that were previously only available in papers. The majority of chapters are self-contained, and all include detailed proofs, a wealth of examples and solved exercises, and a complete reference list. The topics covered include S-Noetherian, S-Artinian, Nonnil-Noetherian, and Strongly Hopfian properties on commutative rings and their transfer to extensions such as polynomial and power series rings, and more. Though primarily intended for readers with a background in commutative rings, modules, polynomials and power series extension rings, the book can also be used as a reference guide to support graduate-level algebra courses, or as a starting point for further research.
Author(s): Ali Benhissi
Publisher: Springer
Year: 2022
Language: English
Pages: 528
City: Cham
Preface
Contents
1 S-Noetherian Modules and Rings
1 S-Noetherian Modules
2 S-Noetherian Rings
3 Modules Over S-Noetherian Rings
4 Polynomials Over S-Noetherian Rings
5 Formal Power Series Over S-Noetherian Rings
6 S-Noetherian Pull Backs
7 S-Noetherian Amalgamations
8 Applications
9 Polynomials and Formal Power Series with Coefficients in a Module
10 S-Noetherian Nagata Rings
11 S-Noetherian t-Nagata Rings
12 Formal Power Series Over a Krull Domain
13 Polynomials Over a Krull Domain
14 Formal Power Series Over a Generalized Krull Domain
Problems
Solutions
References
2 S-Artinian Rings and Modules
1 Saturated Multiplicative Sets of a Ring
2 S-Artinian Rings
3 S-Artinian Modules
4 S-Cofinite Rings and Modules
5 Multiplication Modules
6 S-Artinian and S-Cofinite Idealization
7 Artinian Rings and Modules
8 Application to the Theorem of Generalized Principal Ideal
9 Nagata's Example
Problems
Solutions
References
3 Almost Principal Polynomial Rings
1 Generalities
2 Applications of Anderson-Kwak-Zafrullah Theorem
3 Characterization of A[X] Almost Principal by the Ideals f(X)K[X] A[X]
4 Applications to Extensions of Integral Domains
5 The Almost Principal Ideals of the Form f(X)K[X]A[X] in the Ring A[X]
6 Applications to Semi-Normal Integral Domains
7 The Almost Principal Polynomial Rings and the v-Operation
8 Generators of the Ideal f(X)K[X]A[X]
9 Prestable Ideals
10 When Is the Ideal f(X)K[X] A[X] Maximal?
11 Application to Goldman Rings
Problems
Solutions
References
4 The SFT and t-SFT Rings
1 Krull Dimension of the Formal Power Series Ring
2 Transfer of the SFT Property to the Amalgamation
3 Transfer of the SFT Property to the Polynomial Ring
4 J. Coykendall Example (2002)
5 Transfer of the SFT Property to the Formal Power Series Ring
6 The I-Adic Topology on Rings
7 Infinite Product of Formal Power Series over a Nondiscrete Valuation Domain of Rank One
8 The Zeros of Formal Power Series
9 Infinite Product of Formal Power Series over an Integral Domain
10 Completion of a Metric Space
11 Projective Limit of a Projective System of Rings
12 Construction of the I-Adic Completion of a Ring
13 The I-Adic Completion of a Noetherian Ring
14 The I-Adic Completion of a Valuation Domain
15 Generalities on the t-SFT Domains
16 Transfer of the t-SFT Property to the Polynomial Domain
17 Transfer of the t-SFT Property to the t-Flat Over-Rings
18 Chains Condition in the t-SFT Domains
19 Application to Formal Power Series
Problems
Solutions
References
5 Nonnil-Noetherian Rings
1 Generalities on Nonnil-Noetherian Rings
2 The Nilradical of the Formal Power Series Ring
3 Krull's Dimension of the Formal Power Series Ring over a Nonnil-Noetherian Ring
4 Transfer of the Nonnil-Noetherian Property to the Ring of Formal Power Series
5 Characterization of the Nonnil-Noetherian Property by Formal Power Series
6 Nonnil-S-Noetherian Rings
Problems
Solutions
References
6 Strongly Hopfian, Endo-Noetherian, and Isonoetherian Rings
1 Generalities on the Strongly Hopfian Rings
2 The Zero Dimensional Rings
3 Auto-Injective Rings
4 Product of Zero Dimensional Rings
5 Injection in a Zero Dimensional Ring
6 Polynomials Over a Strongly Hopfian Ring
7 Formal Power Series Over a Strongly Hopfian Bounded Ring
8 Formal Power Series Over a Chained Ring
9 Formal Power Series Over a Ring with Nonzero Characteristic
10 Study of an Example
11 Other Criteria for the Strong Hopfianity
12 Power Series Armendariz Rings
13 Endo-Noetherian Rings
14 Polynomials and Formal Power Series Over an Endo-Noetherian Ring
15 Polynomials and Formal Power Series Over a PF-Ring
16 Isonoetherian Valuation Domains
17 Isonoetherian Modules
18 u-Isonoetherian Extensions of Rings
19 Isonoetherian Rings of the Form A+XB[X]/A+XB[[X]]
20 Isonoetherian Nagata's Idealization Ring
Problems
Solutions
References
Index
