Commutative Algebra 1st Edition by Aron Simis ISBN 3110617072 9783110617078

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Product details: 

ISBN 10: 3110617072

ISBN 13: 9783110617078

Author: Aron Simis

This unique book on commutative algebra is divided into two parts in order to facilitate its use in several types of courses. The first introductory part covers the basic theory, connections with algebraic geometry, computational aspects, and extensions to module theory. The more advanced second part covers material such as associated primes and primary decomposition, local rings, M-sequences and Cohen-Macaulay modules, and homological methods.

Commutative Algebra 1st Table of contents: 

1. Introduction to Commutative Algebra

   • Basic Definitions and Notation
   • Motivation and Applications
   • Overview of Key Concepts in Commutative Algebra

2. Rings and Ideals

   • Definitions of Rings, Subrings, and Fields
   • Ideals and Quotient Rings
   • Operations on Ideals
   • Prime and Maximal Ideals

3. Ring Homomorphisms and Isomorphisms

   • Definitions and Properties
   • Kernel and Image
   • First Isomorphism Theorem

4. Modules over Rings

   • Definition and Examples of Modules
   • Submodules and Quotient Modules
   • Module Homomorphisms and Isomorphisms
   • Exact Sequences

5. Primary Decomposition and Noetherian Rings

   • Noetherian Rings and Modules
   • Chain Conditions on Ideals and Modules
   • Primary Decomposition of Modules

6. Localization

   • Localization of Rings and Modules
   • Properties and Applications
   • Localization at a Prime Ideal

7. Dimension Theory

   • Krull Dimension of Rings and Modules
   • Chains of Prime Ideals
   • The Dimension of Polynomial Rings

8. Integral Elements and Integrality

   • Integral Extensions and Integral Closure
   • Integrality of Elements and Rings
   • Applications to Algebraic Geometry

9. Complete Local Rings

   • Definition and Properties of Complete Local Rings
   • The Artin-Rees Lemma
   • Applications to Commutative Algebra

10. Noetherian Local Rings and Regular Sequences

    • Regular Sequences and Depth
    • The Auslander-Buchsbaum Formula
    • Applications to Local Rings

11. Associated Primes and Homological Methods

    • Associated Primes of Modules
    • Depth and Cohen-Macaulay Rings
    • Homological Tools in Commutative Algebra

12. Applications and Further Topics

    • The Relationship Between Algebra and Geometry
    • Hilbert’s Nullstellensatz and Algebraic Geometry
    • Homological Methods in Commutative Algebra

13. Appendix

    • Useful Lemmas and Theorems
    • Tables and Reference Material

14. Index

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Aron Simis,Commutative Algebra