A Quick Review of Commutative Algebra

Lectures on Commutative Algebra

Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology, Bombay
Powai, Mumbai 400076 India
E-mail: srg@math.iitb.ac.in

Version 1.1, July 2006

Description

These notes are based on the lectures given at the NBHM sponsored Annual Foundational School-II held at Bhaskaracharya Pratishthana, Pune and the Department of Mathematics, University of Pune in June 2006. Like the earlier notes, A Quick Review of Commutative Algebra, they attempt to give a rapid review of the rudiments of classical commutative algebra. However, some of the basic material is covered in greater detail and some new material is added. Exercises are collated at the end of each chapter. Further material on Dedekind domains, as a continuation of the topics covered in Chapter 4, can be found in the Kiel notes.

Contents

1 Rings and Modules 3
1.1 Ideals and Radicals 3
1.2 Polynomial rings and Localization of rings 8
1.3 Modules 11
1.4 Zariski Tolpology 12
Exercises 14
2 Noetherian Rings 17
2.1 Noetherian Rings and Modules 17
2.2 Primary Decomposition of Ideals 19
2.3 Artinian Rings and Modules 23
2.4 Krull's Principal Ideal Theorem 27
Exercises 14
3 Integral Extensions 32
3.1 Integral Extensions 32
3.2 Noether Normalization 35
3.3 Finiteness of Integral Closure 38
Exercises 42
4 Dedekind Domains 44
4.1 Dedekind Domains 45
4.2 Extensions of Primes 50
Exercises 42
A Appendix: Primary Decomposition of Modules 55
A.1 Associated Primes of Modules 55
A.2 Primary Decomposition of Modules 58
Exercises 62
References 63

1	Rings and Modules	3
	1.1 Ideals and Radicals	3
	1.2 Polynomial rings and Localization of rings	8
	1.3 Modules	11
	1.4 Zariski Tolpology	12
	Exercises	14
2	Noetherian Rings	17
	2.1 Noetherian Rings and Modules	17
	2.2 Primary Decomposition of Ideals	19
	2.3 Artinian Rings and Modules	23
	2.4 Krull's Principal Ideal Theorem	27
	Exercises	14
3	Integral Extensions	32
	3.1 Integral Extensions	32
	3.2 Noether Normalization	35
	3.3 Finiteness of Integral Closure	38
	Exercises	42
4	Dedekind Domains	44
	4.1 Dedekind Domains	45
	4.2 Extensions of Primes	50
	Exercises	42
A	Appendix: Primary Decomposition of Modules	55
	A.1 Associated Primes of Modules	55
	A.2 Primary Decomposition of Modules	58
	Exercises	62
	References	63

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