**MATH 521, Spring 2008 **

10:00-10:50, MWF, DU 240

**Professor John Beachy**,
Watson 355, 753-6753

**Office Hours**:
11:00-11:50 MWF
(in Watson 355),
or by appointment

SYLLABUS

**COURSE:**
ALGEBRAIC STRUCTURES II (3)
Ring theory including the Artin-Wedderburn theorem, the Jacobson radical.
Commutative algebra, Noetherian rings, and Dedekind domains.

**PREREQUISITE:**
MATH 520 or consent of department

**TEXT:**
**Introductory Lectures on Rings and Modules**,
Cambridge University Press,
and supplementary lecture notes on commutative rings.
Recommended reference: **Algebra**, by Hungerford.

**SYLLABUS:**
These are the topics listed for the comprehensive exam in algebra.

Rings and Modules:
Modules, simplicity, semisimplicity, chain conditions, tensor products,
Jacobson radical, density theorem, Wedderburn-Artin theorem,
finitely generated modules over a principal ideal domain,
canonical forms.

Unique factorization, Euclidean domains, principal ideal domains,
polynomial rings, maximal, prime, and primary ideals,
Noetherian rings, Hilbert basis theorem, Lasker-Noether decomposition,
integral elements, integral closure, fractional ideals, Dedekind domains.

To prepare for these topics,
I plan to cover the following sections of the text.
(Chapter 5 is a set of supplementary notes available on the web,
and is not included in the published version of the book).
The emphasis of the book is on noncommutative rings,
and so Chapter 4 covers the representation theory of finite groups
as an application of the general theory developed in the previous chapters.

Chapter 1: Rings (1.1-1.5);
Chapter 2: Modules (2.1-2.7);
Chapter 3: Structure of noncommutative rings (3.1-3.3);
Chapter 5: Ideal theory of commutative rings
(5.1 Dedekind domains, 5.2 Integral extensions,
5.3 Primary decomposition, 5.4 Noetherian rings)

TENTATIVE SCHEDULE OF LECTURES

MONDAY WEDNESDAY FRIDAY M Tu W Th F Week of JAN 2008 1/14 1.1 1.2 1.2 14 15 16 17 18 1/21 HOLIDAY 1.3 1.3 21 22 23 24 25 1/28 1.4 1.5 1.5 28 29 39 31 1 2/4 2.1 2.1 2.1 FEB 4 5 6 7 8 2/11 2.2 2.2 2.3 11 12 13 14 15 2/18 2.3 2.4 2.4 18 19 20 21 22 2/25 2.5 2.5 2.6 25 26 27 28 29 3/3 MIDTERM 2.6 2.6 MAR 3 4 5 5 7 SPRING BREAK 10 11 12 13 14 3/17 2.7 2.7 2.7 17 18 19 20 21 3/24 3.1 3.1 3.1 24 25 26 27 28 3/31 3.2 3.2 3.3 APR 31 1 2 3 4 4/7 3.3 3.3 5.1 7 8 9 10 11 4/14 5.1 5.1 5.2 14 15 16 17 18 4/21 5.2 5.3 5.3 21 22 23 24 25 4/28 5.4 5.4 READING DAY 28 29 30 1 2 5/5 FINAL MAY 5 6 7 8 9 FINAL EXAM: Monday, May 5, 10:00-11:50 a.m.

**GRADING:**
Semester grades will be based on 500 points:
100 points for the midterm exam;
200 points for homework; and
200 points for the final exam.

**MIDTERM:**
The midterm exam is tentatively scheduled for Monday, March 3.

**FINAL:**
The final exam is scheduled for
Monday, May 5, 10:00-10:50 a.m.