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Atiyah-Macdonald Solutions Solutions to the Problems in Introduction to Commutative Algebra by M. F. Atiyah and I. G. MacDonald J. David Taylor October 20, 2018 Contents 1 Rings and Ideals 7 1.1........................................................7 1.2........................................................7 1.3........................................................8 1.4........................................................8 1.5........................................................8 1.6........................................................9 1.7........................................................ 10 1.8........................................................ 10 1.9........................................................ 10 1.10........................................................ 10 1.11........................................................ 10 1.12........................................................ 11 1.13........................................................ 11 1.14........................................................ 11 1.15........................................................ 12 1.16........................................................ 12 1.17........................................................ 14 1.18........................................................ 15 1.19........................................................ 15 1.20........................................................ 15 1.21........................................................ 16 1.22........................................................ 17 1.23........................................................ 18 1.24........................................................ 18 1 CONTENTS 2 1.25........................................................ 19 1.26........................................................ 20 1.27........................................................ 20 1.28........................................................ 20 2 Modules 22 2.1........................................................ 22 2.2........................................................ 22 2.3........................................................ 23 2.4........................................................ 23 2.5........................................................ 24 2.6........................................................ 24 2.7........................................................ 25 2.8........................................................ 25 2.9........................................................ 25 2.10........................................................ 26 2.11........................................................ 26 2.12........................................................ 27 2.13........................................................ 27 2.14........................................................ 27 2.15........................................................ 27 2.16........................................................ 28 2.17........................................................ 29 2.18........................................................ 29 2.19........................................................ 29 2.20........................................................ 30 2.21........................................................ 30 2.22........................................................ 31 2.23........................................................ 31 2.24........................................................ 31 2.25........................................................ 31 2.26........................................................ 32 2.27 Incomplete................................................. 32 2.28........................................................ 33 3 Rings and Modules of Fractions 34 3.1 Current................................................... 34 3.2........................................................ 34 CONTENTS 3 3.3........................................................ 34 3.4........................................................ 35 3.5........................................................ 35 3.6........................................................ 35 3.7........................................................ 35 3.8........................................................ 36 3.9........................................................ 36 3.10........................................................ 37 3.11........................................................ 37 3.12........................................................ 38 3.13........................................................ 39 3.14........................................................ 39 3.15........................................................ 39 3.16........................................................ 39 3.17........................................................ 40 3.18........................................................ 40 3.19........................................................ 40 3.20........................................................ 41 3.21........................................................ 42 3.22........................................................ 44 3.23........................................................ 44 3.24........................................................ 44 3.25........................................................ 45 3.26........................................................ 45 4 Primary Decomposition 46 4.1........................................................ 46 4.2........................................................ 46 4.3........................................................ 46 4.4........................................................ 46 4.5........................................................ 46 4.6........................................................ 46 4.7........................................................ 47 4.8........................................................ 47 4.9........................................................ 47 4.10........................................................ 48 4.11........................................................ 48 CONTENTS 4 4.12........................................................ 49 4.13........................................................ 49 4.14........................................................ 50 4.15........................................................ 50 4.16........................................................ 50 4.17........................................................ 50 4.18........................................................ 50 4.19........................................................ 51 4.20........................................................ 51 4.21........................................................ 52 4.22........................................................ 52 4.23........................................................ 53 5 Integral Dependence and Valuations 54 5.1........................................................ 54 5.2........................................................ 54 5.3........................................................ 54 5.4........................................................ 54 5.5........................................................ 55 5.6........................................................ 55 5.7........................................................ 55 5.8........................................................ 55 5.9........................................................ 56 5.10........................................................ 56 5.11........................................................ 56 5.12........................................................ 56 5.13........................................................ 57 5.14........................................................ 57 5.15........................................................ 57 5.16........................................................ 57 5.17........................................................ 58 5.18........................................................ 58 5.19 Undone................................................... 58 5.20........................................................ 58 5.21........................................................ 58 5.22 Undone................................................... 59 5.23........................................................ 59 CONTENTS 5 5.24........................................................ 59 5.25........................................................ 60 5.26........................................................ 60 5.27........................................................ 61 5.28........................................................ 61 5.29........................................................ 62 5.30........................................................ 62 5.31........................................................ 62 5.32 Undone................................................... 63 5.33 Undone................................................... 63 5.34 Undone................................................... 63 5.35 Undone................................................... 63 6 Chain Conditions 64 6.1........................................................ 64 6.2........................................................ 64 6.3........................................................ 64 6.4........................................................ 65 6.5........................................................ 65 6.6.......................................................
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