Commutative Ring Theory

In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings. More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored. The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject. Exercises are provided at the end of each section and solutions or hints to some of them are given at the end of the book.

Preface; Introduction; Conventions and terminology; 1. Commutative rings and modules; 2. prime ideals; 3. Properties of extension rings; 4. Valuation rings; 5. Dimension theory; 6. Regular sequences; 7. Regular rings; 8. Flatness revisited; 9. Derivations; 10. I-smoothness; 11. Applications of complete local rings; Appendices; References; Index.

In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings.