MODERN ALGEBRA AND APPLICATIONS covers some basic results pertaining to the algebraic structures like groups, rings and modules. It also contains some advanced topics like associated prime ideals, primary decomposition, skew polynomial rings and their applications. This area is now an active area of research. The book shall be useful for undergraduates and post graduates (research scholars).
Preface vii
Notations xiii
Introduction xv
1 Preliminaries 1 (36)
1.1 Set Theory and Groups 2 (6)
1.2 Rings 8 (5)
1.3 Ideals 13 (9)
1.4 Divisibility 22 (3)
1.5 Euclidean Domain 25 (1)
1.6 Principal Ideal Domain 26 (4)
1.7 Modules 30 (6)
1.8 Exercises 36 (1)
2 Polynomial Rings 37 (10)
2.1 Ring of Polynomials 37 (4)
2.2 Content of Polynomial and Primitive 41 (2)
Polynomial
2.3 Ring of Polynomials over a UFD 43 (1)
2.4 Eisentein's Irreducible Criterion 44 (2)
2.5 Exercises 46 (1)
3 Modules with Chain Conditions 47 (42)
3.1 Chain Conditions: Artinian Modules, 48 (9)
Noetherian Modules
3.2 Modules of Finite Length 57 (4)
3.3 Artinian Rings 61 (3)
3.4 Noetherian Rings 64 (8)
3.5 Radicals: Nil Radical, Jacobson 72 (8)
Radical
3.6 Radical of an Artinian Ring 80 (7)
3.7 Exercises 87 (2)
4 Skew Polynomial Rings 89 (51)
4.1 Endomorphisms and Derivations 90 (4)
4.2 Skew Polynomial Rings of Endomorphism 94 (7)
Type
4.3 Skew Polynomial Rings of Derivation 101 (5)
Type
4.4 Skew Laurent Rings 106 (8)
4.5 General Skew Polynomial Rings 114 (6)
4.6 Skew Polynomial Rings (particular 120 (20)
cases)
5 Primary Decomposition 140 (32)
5.1 Associated Prime Ideals 141 (4)
5.2 Primary Decomposition of Modules 145 (2)
5.3 Primary Decomposition in Noetherian 147 (3)
Rings
5.4 Krull Dimension 150 (2)
5.5 Krull Dimension of Polynomial and 152 (4)
Skew Polynomial Rings
5.6 Ideal Krull-symmetry of Polynomial 156 (6)
Rings
5.7 Primary Decomposition in 162 (2)
Non-Noetherian Rings
5.8 Primary Decomposition in Rings with 164 (4)
Quotient Rings
5.9 Artinian Embedding 168 (4)
6 Primary Decomposition of Skew Polynomial 172 (22)
Rings
6.1 Associated Prime Ideals of Skew 172 (4)
Polynomial Rings of Automorphism Type
6.2 Associated Primes Ideals of Skew 176 (1)
Laurent Rings
6.3 Associated Primes Ideals of Skew 177 (2)
Polynomial Rings of Derivation Type
6.4 Completely Prime Ideals of Polynomial 179 (2)
Rings
6.5 Strongly Prime Ideals of Polynomial 181 (1)
Rings
6.6 Transparent Rings and their Extensions 182 (4)
6.7 Transparent Skew Polynomial Rings 186 (8)
(Special cases)
7 Applications of Skew polynomial Rings 194 (33)
7.1 Coding Theory 194 (7)
7.2 Codes over Skew Polynomial Rings 201 (4)
7.3 The Length of θ-Code 205 (5)
7.4 Skew Polynomial Rings for an Analysis 210 (6)
of Control Systems
7.5 Ordinary Differential Equations with 216 (6)
Skew Polynomial Rings
7.6 General Derivations and Differential 222 (5)
Operators
References 227 (8)
Index 235
新书报道