This book grew out of a course of lectures given to third year
undergraduates at Oxford University and it has the modest aim of
producing a rapid introduction to the subject. It is designed to be read
by students who have had a first elementary course in general algebra.
On the other hand, it is not intended as a substitute for the more
voluminous tracts such as Zariski-Samuel or Bourbaki. We have
concentrated on certain central topics, and large areas, such as field
theory, are not touched. In content we cover rather more ground than
Northcott and our treatment is substantially different in that,
following the modern trend, we put more emphasis on modules and
localization.
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This is a comprehensive review of commutative algebra, from localization
and primary decomposition through dimension theory, homological
methods, free resolutions and duality, emphasizing the origins of the
ideas and their connections with other parts of mathematics. The book
gives a concise treatment of Grobner basis theory and the constructive
methods in commutative algebra and algebraic geometry that flow from it.
Many exercises included.