Commutative Algebra (London Mathematical Society Lecture Note Series, Series Number 5)

This Introduction To Commutative Algebra Gives An Account Of Some General Properties Of Rings And Modules, With Their Applications To Number Theory And Geometry. It Assumes Only That The Reader Has Completed An Undergraduate Algebra Course. The Fresh Approach And Simplicity Of Proof Enable A Large Amount Of Material To Be Covered; Exercises And Examples Are Included Throughout The Notes. [by] J. T. Knight. Cover; Title; Copyright; Contents; 1. Preliminaries; 1. 1 INTRODUCTION; 1. 2 DEFINITIONS AND RECAPITULATIONS; 1.3 MODULES; 2. Flatness; 2.1 PROJECTIVE MODULES; 2.2 FLAT MODULES; 2.3 FAITHFULLY FLAT MODULES; 3. Fractions; 3.1 RINGS, MODULES AND ALGEBRAS OF FRACTIONS; 3.2 LOCALISATION; 3.3 PROJECTIVE MODULES AND LOCALISATION; 3.4 SUBMODULES OF FRACTION ALGEBRAS; 4. Supporting and associated prime ideals; 4.1 LENGTHS AND RANKS OF MODULES; 4.2 THE SUPPORT OF A MODULE; 4.3 PRIME IDEALS ASSOCIATED TO A MODULE; 5. Integers; 5.1 DEFINITION OF INTEGERS; 5.2 INTEGERS AND PRIME IDEALS 6. Some geometrical resultsAPPLICATION TO ALGEBRAIC CLOSURES OF FIELDS; 7. Valuation rings; 7.1 ORDERED GROUPS; 7.2 VALUATION RINGS; 7.3 EXTENSION THEOREMS; 7.4 AN APPLICATION; EXAMPLES ON GENERAL VALUATIONS; 8. Prüfer and Dedekind rings; 9. General exercises; Appendix 1: A NOTE ON CATEGORIES, FUNCTORS AND NATURAL TRANSFORMATIONS; Appendix 2: THE CONSTRUCTIBLE TOPOLOGY; Bibliography; Index of notation; Index of terms