Undergraduate Commutative Algebra

For those looking for an introduction to the area of commutative algebra, this book opens all the right doors and provides a clarity of understanding that all will welcome.

Miles Reid (Author)

9780521452557, Cambridge University Press

Hardback, published 30 November 1995

168 pages, 11 b/w illus.
23.5 x 15.6 x 1.6 cm, 0.371 kg

'Besides the usual topics … there are some welcome geometrical illustrations, as well as some homespun philosophy.' Mathematica

Commutative algebra is at the crossroads of algebra, number theory and algebraic geometry. This textbook is affordable and clearly illustrated, and is intended for advanced undergraduate or beginning graduate students with some previous experience of rings and fields. Alongside standard algebraic notions such as generators of modules and the ascending chain condition, the book develops in detail the geometric view of a commutative ring as the ring of functions on a space. The starting point is the Nullstellensatz, which provides a close link between the geometry of a variety V and the algebra of its coordinate ring A=k[V]; however, many of the geometric ideas arising from varieties apply also to fairly general rings. The final chapter relates the material of the book to more advanced topics in commutative algebra and algebraic geometry. It includes an account of some famous 'pathological' examples of Akizuki and Nagata, and a brief but thought-provoking essay on the changing position of abstract algebra in today's world.

Hello!
1. Basics
2. Modules
3. Noetherian rings
4. Finite extensions and Noether normalisation
5. The nullstellensatz and spec A
6. Rings of fractions S-1A and localisation
7. Primary decomposition
8. DVRs and normal integral domains
9. Goodbye!
Bibliography.

Subject Areas: Algebra [PBF]