Syllabus

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

Amazon logo Reid, Miles. Undergraduate Commutative Algebra: London Mathematical Society Student Texts. Cambridge, UK: Cambridge University Press, April 26, 1996. ISBN: 9780521458894.

Amazon logo Atiyah, Michael, and Ian Macdonald. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1994. ISBN: 9780201407518.

Recommended

Amazon logo 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