Contents Preface Chapter 1 Basic Concepts 1 1.1 Rings and modules 1 1.1.1 Rings and their homomorphisms 1 1.1.2 Modules and their homomorphisms 4 1.1.3 Free, projective, injective and at modules 9 1.1.4 Covers and envelopes 11 1.2 Some constructions of rings 12 1.2.1 Formal power series and polynomials 12 1.2.2 Matrices 13 1.2.3 Morita context and formal triangular matrices 16 1.2.4 Direct products 17 1.2.5 Group rings 18 1.2.6 Localizations of commutative rings 19 Chapter 2 Noncommutative Rings 21 2.1 Asymmetry of noncommutative rings 21 2.1.1 Artinian ring and noetherian ring 21 2.1.2 Bass ring 22 2.1.3 B.ezout ring 23 2.1.4 Bounded ring 25 2.1.5 Coherent ring 26 2.1.6 Cononsingular ring 29 2.1.7 Continuous, quasi-continuous and CS ring 30 2.1.8 Distributive ring 31 2.1.9 Dual ring and quasi-dual ring 32 2.1.10 Duo ring 33 2.1.11 Finitely embedded ring 34 2.1.12 F-injective ring, FP-injective ring 35 2.1.13 Free ideal ring 37 2.1.14 FS ring and PS ring 38 2.1.15 Goldie ring 40 2.1.16 GP-injective ring 40 2.1.17 Harada ring and quasi-Harada ring 43 2.1.18 Hereditary ring and semihereditary ring 44 2.1.19 IN ring 45 2.1.20 Kasch ring and strong Kasch ring 46 2.1.21 McCoy ring 47 2.1.22 Min-injective ring and strong min-injective ring 48 2.1.23 Minsymmetric ring 49 2.1.24 Morphic ring 49 2.1.25 Nonsingular ring 51 2.1.26 Ore ring 51 2.1.27 PF ring 52 2.1.28 Primitive ring 53 2.1.29 P-injective ring 55 2.1.30 Principally quasi-Baer ring 55 2.1.31 Repetitive ring 56 2.1.32 RF ring 57 2.1.33 Self-injective ring 57 2.1.34 Serial ring 59 2.1.35 Simple-injective ring 60 2.1.36 Soc-injective ring 61 2.1.37 Strong rank condition 62 2.1.38 Strongly prime ring 63 2.1.39 V ring 64 2.1.40 Weakly regular ring 65 2.1.41 Zip ring 65 2.2 Weak commutativity 67 2.2.1 Abelian ring 67 2.2.2 Compressible ring 67 2.2.3 Directly-nite ring 68 2.2.4 Duo ring 69 2.2.5 Reversible ring 69 2.2.6 Semi-commutative ring 70 2.2.7 Symmetric ring 72 2.3 Remainders 74 2.3.1 IBN ring 74 2.3.2 McCoy ring 75 2.3.3 Ore ring 76 2.3.4 Stably-nite ring 76 Chapter 3 Hierarchies 78 3.1 Domain 78 3.2 Chain condition and-niteness condition 82 3.3 Armendariz ring 92 3.4 Weak commutativity 96 3.5 Von Neumann regularity 103 3.6 Baer ring 113 3.7 Injectivity 117 3.8 Continuity 128 3.9 Morphic ring 133 3.10 Clean ring 137 3.11 Hereditary ring 151 3.12 Primitive ring 154 3.13 V ring 161 3.14 Quasi-Frobenius ring 166 3.15 Involutive ring 172 3.15.1 Baer *-ring 172 3.15.2 *-clean ring 173 3.15.3 *-regular ring 175 3.15.4 Symmetric *-ring 177 Chapter 4 Extensions 179 4.1 Matrix ring 179 4.1.1 Armendariz ring 179 4.1.2 Clean ring 181 4.1.3 Dual ring 182 4.1.4 Goldie ring 183 4.1.5 Continuity and injectivity 184 4.1.6 McCoy ring 188 4.1.7 Morphic ring 188 4.1.8 Involutive ring 192 4.1.9 Zip ring 193 4.2 Polynomial ring 193 4.2.1 Armendariz ring 193 4.2.2 Baer ring and Rickart ring 193 4.2.3 Clean ring 195 4.2.4 Coherent ring 197 4.2.5 Goldie ring 199 4.2.6 McCoy ring 199 4.2.7 Morphic ring 200 4.2.8 Ore domain 201 4.2.9 Weak commutativity 202 4.2.10 Zip ring 203 4.3 Group ring 203 4.3.1 Baer ring and Rickart ring 203 4.3.2 Clean ring 204 4.3.3 Finite dimensional ring 205 4.3.4 Morphic ring 206 4.3.5 Prime ring and primitive ring 206 4.3.6 Von Neumann regular ring 207 4.3.7 Zip ring 208 4.4 Subring of direct product 209 Bibliography 216 Index 233
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