Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Description
In this course students will learn about Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, DVRs, filtrations, length, Artin rings, Hilbert polynomials, tensor products, and dimension theory.
Prerequisites
Algebra I (18.701) and Algebra II (18.702)
Textbooks
Required
Reid, Miles. Undergraduate Commutative Algebra: London Mathematical Society Student Texts. Cambridge, UK: Cambridge University Press, April 26, 1996. ISBN: 9780521458894.
Atiyah, Michael, and Ian Macdonald. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1994. ISBN: 9780201407518.
Recommended
Eisenbud, David. Commutative Algebra: With a View Toward Algebraic Geometry. New York, NY: Springer-Verlag, 1999. ISBN: 9780387942698.
Grading
Grades in this class are based nearly entirely on the problem sets. In addition, each student will present (at least) one problem in class. There will be no exams.
Homework
Problem sets are due in class each Thursday of the week after they are assigned (in other words, 7 or 9 days later). On occasion, a late problem set will be accepted provided that you state (1) a good reason why you need the extra time and (2) the date when the set will be turned in. Collaboration is permitted, indeed encouraged, provided that you think through each problem on your own and write it up in your own words. Problem sets will be graded in part on the quality of the write-up.
Calendar
| SES # | TOPICS |
|---|---|
| 1-3 | Rings and ideals |
| 4-6 | Modules |
| 7-8 | Integral dependence |
| 9-13 | Localization |
| 14 | Primary decomposition |
| 15-19 | Dedekind domains |
| 20-22 | Dimension theory |
| 23 | Tensor product |
| 24-26 | Length |
