Commutative Algebra is the study of commutative rings, and their modules and ideals. This theory has developed over the last 150 years not just as an area of algebra considered for its own sake, but as a tool in the study of two enormously important branches of mathematics: algebraic geometry and algebraic number theory. The unification which results, where the same underlying algebraic structures arise both in geometry and in number theory, has been one of the crowning glories of twentieth century mathematics and still plays an absolutely fundamental role in current work in both these fields.
One simple example of this unification will be familiar already to anyone who has noticed the strong parallels between the ring Z (a Euclidean Domain and hence also a Unique Factorization Domain) and the ring F[X] of polynomials over a field (which has both the same properties). More generally, the rings of algebraic integers which have been studied since the 19th century to solve problems in number theory have parallels in rings of functions on curves in geometry.
While self-contained, this course will also serve as a useful introduction to either algebraic geometry or algebraic number theory.
Topics: Gröbner bases, modules, localization, integral closure, primary decomposition, valuations and dimension.
This course will give the student a solid grounding in commutative algebra which is used in both algebraic geometry and number theory.
M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra. Addison-Wesley 1969; reprinted by Perseus 2000. [QA251.3.A8]
D. Eisenbud, Commutative algebra with a view toward algebraic geometry. Springer 1995. [QA251.3.E4]
M. Reid, Undergraduate Commutative Algebra. CUP 1995. [QA251.3.R3]
R.Y. Sharp, Steps in Commutative Algebra (2nd ed.) CUP 2000. [QA251.3.S4]
O. Zariski and P. Samuel, Commutative Algebra, (vols I and II). Springer 1975-6. [QA251.3.Z2]