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C
ommutative
a
lgebra
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C
ommutative
a
lgebra
An Introduction
J. W
illiam
H
offman
(Louisiana State University)
(Chinese Academy of Sciences)
X
iaoHong
J
ia
(Southeast Missouri State University)
H
aoHao
W
ang
M
ercury
L
earning and
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nforMation
Dulles, Virginia
Boston, Massachusetts
New Delhi
Copyright ©2016 by
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ercury
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earning
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Original title and copyright:
Essentials of Commutative Algebra.
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ercury
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J. W. Hoffman, X. Jia, H. Wang.
Commutative Algebra. An Introduction.
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C
ontents
Preface �½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½�½
vii
1. Introduction ...........................................................................1
2. Rings and Ideals ....................................................................9
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Rings and Ideals .......................................................................................9
Localization of a Ring ............................................................................19
Ideals in a Polynomial Ring ..................................................................27
Gröbner Basis of an Ideal ......................................................................32
Elimination and Extension ...................................................................38
Implicitization ........................................................................................45
Schemes ..................................................................................................48
Gröbner Basis Applications ..................................................................53
2.8.1 Solving Systems of Equations .....................................................54
2.8.2 Orthogonal Projection ................................................................56
2.8.3 Poncelet’s Algebraic Correspondence ........................................58
Modules ..................................................................................................63
Exact Sequences and Commutative Diagrams ....................................67
Projective and Injective Modules .........................................................71
Tensor Product of Modules ..................................................................75
Flatness ...................................................................................................78
Localization ............................................................................................80
Local Property ........................................................................................82
Associated Primes ..................................................................................85
Primary Decomposition of Modules ...................................................89
Modules of Finite Length ......................................................................92
3. Modules ...............................................................................63
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10

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