Department of Mathematical Sciences
Northern Illinois University

## MATH 521, Spring 2008

#### 10:00-10:50, M W F, DU 240

Professor John Beachy, Watson 355, 753-6753

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

email: beachy@math.niu.edu | My faculty homepage | My personal homepage

Some exams from previous semesters Midterm and Final from 1993 | Midterm and Final from 1999 | Final solutions from 1999
Midterm from 2005 (this one looks easier than it should be--we've covered more material) | Final from 2005

## 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 (62 pages)

1.1 Basic definitions and examples
1.2 Ring homomorphisms
1.3 Localization of integral domains
1.4 Unique factorization

CHAPTER 2: MODULES (74 pages)

2.1 Basic definitions and examples
2.2 Direct sums and products
2.3 Semisimple modules
2.4 Chain conditions
2.5 Modules with finite length
2.6 Tensor products
2.7 Modules over principal ideal domains

CHAPTER 3: STRUCTURE OF NONCOMMUTATIVE RINGS (34 pages)

3.1 Prime and primitive ideals
3.3 Semisimple Artinian rings

CHAPTER 5: IDEAL THEORY OF COMMUTATIVE RINGS (36 pages)

5.1 Dedekind domains
5.2 Integral extensions
5.3 Primary decomposition
5.4 Noetherian rings

## TENTATIVE SCHEDULE OF LECTURES

Chapter 1, Rings: 3 weeks
Chapter 2, Modules: 6 weeks
Chapter 3, Noncommutative Rings: 3 weeks
Chapter 5, Commutative Rings: 3 weeks
```
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.
```

100 points for the midterm exam
200 points for homework
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.

## ASSIGNMENTS

```
DUE   SECTION    PROBLEMS

Homework #8:

3.1       #2  (Correct the mistake in the question, then
give the solution.)
3.2       #3,4
3.3       #2,5

```

## CLASS NOTES

Class notes, a rough draft, in pdf format

## REFERENCES

Algebra, Hungerford
Basic Algebra I, II, Jacobson
Abstract Algebra, Dummit and Foote
An Introduction to Ring Theory, Cohn

## RESOURCES ON THE WEB

Complete books online:
Elements of Abstract and Linear Algebra, by Edwin Connell
Abstract Algebra: The Basic Graduate Year, by Robert Ash
A Course in Commutative Algebra, by Robert Ash

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