Introduction to Commutative Algebra

Fall 2012

INSTRUCTOR:

Sarah Glaz

glaz@math.uconn.edux (click on link and remove end x)
Office: MSB 202
Phone: (860) 486 9153
Office Hours: T, Th 11:15 - 12:15 and by appointment

CLASS:

T, Th 12:30 - 1:45, MSB 117

BOOKS:

Main Textbook:  Introduction to Commutative Algebra, by M.F. Atiyah & I.G. MacDonald, Academic Press, Inc.
(May be ordered new or used from amazon.com, or other reliable online used book sites like alibris.com, abebooks.com, or strandbooks.com)
Other Texts (you need not purchase):
Commutative Algebra, by Bourbaki
Commutative Coherent Rings, by Sarah Glaz
Multiplicative Ideal Theory,
by Robert Gilmer
Commutative Rings with Zero-Divisors, by James Huckaba
Commutative Rings, by Irving Kaplansky
Commutative Ring Theory, by Hideyuki Matsumura

COURSE OUTLINE:

Commutative Algebra, the study of commutative rings and their modules, emerged as a definite area of mathematics at the beginning of the twentieth century. Its origins lie in the works of eminent mathematicians such as Kronecker, Dedekind, Hilbert and Emy Noether who sought to develop a solid foundation for Number Theory. Later, the field was enriched by its relation to modern Algebraic Geometry, Topology, Homological Algebra, and Combinatorics. Today Commutative Algebra is a deep and beautiful area of study in its own right, which both draws on, and is applicable to, all the disciplines that contributed to its development. In this course we will study the fundamental notions and methods of research of commutative algebra. Topics will include: basic module and ideal notions and constructions (such as prime ideals, zero-divisors, localizations, primary decomposition, integral dependence, completions, and dimension theory), special types of rings (such as valuation rings, Krull domains, Noetherian rings, Artinian rings, and coherent rings), and homological algebra aspects of commutative algebra (such as projectivity, flatness, grade, and factorization properties). Other topics may be introduced as time allows. We anticipate covering most of  Chapters 1 - 11 from the main textbook, supplemented from the other texts mentioned above.

Grading would be based on a homework grade (50%), and a semester project grade (50%). There will be 3 - 5 homework assignments, each comprising of a number of exercises. Students are encouraged to discuss the homework assignments with each other, but need to formulate, write, and hand in their own individual solutions. The semester project will be on a topic chosen by the instructor in collaboration with each student. Each students will hand in a typed report (5 - 10 pages) and give a class presentation on his/her project. Topics will be assigned a few weeks into the semester. Reports and presentations will begin in November.

SPECIAL DAYS:

First Day of Classes: Monday, August 27
Labor Day (no classes): Monday, September 3
No class: Tuesday, September 18
Thanksgiving Break: Sunday, November 18 - Sunday, November 25
Last Day of Classes: Friday, December 7

HOMEWORK  AND PRESENTATIONS

Homework 1: Chapter 1: Exercises 1,2,6,7,11,12
Solutions to HW1 by Ryan Pellico

Homework 2: Chapter 2: Exercises 1,2,3,4,5,6
Chapter 1: Exercises 15,16
Solutions to HW2 by Mingfeng Zhao

Schedule of Presentations and Reports