THE UNIVERSITY OF WESTERN ONTARIO

LONDON CANADA

DEPARTMENT OF MATHEMATICS

__Mathematics 4123A/9023A: Rings and Modules__

**Fall 2014**

**
**

**INSTRUCTOR:** Lex Renner

**TEXTBOOK:** Rings Modules and Linear Algebra by B. Hartley and T. O. Hawkes, Chapman and Hall,
http://www.unalmed.edu.co/~mmtoro/doc/hartleyhawkes.pdf

**PREREQUISITES:** Mathematics 3020 A/B, Mathematics 3120 A/B, Mathematics 3121 A/B (formerly Mathematics 2121 A/B)
or permission from the instructor.

**COURSE OUTLINE:**
Commutative rings, ring homomorphisms and quotient rings, ideals, rings of fractions, the Chinese remainder theorem;
Euclidean domains, principal ideal domains, unique factorization domains; polynomial rings over fields; modules, direct
sums of modules, free modules; modules over a principal ideal domain, the rational canonical form, the Jordan canonical form.

**EVALUATION OF STUDENT PERFORMANCE:**

Assignments---worth 40%

In-class Midterm #1---October 21, 2014----worth 10%

In-class Midterm #2---November 20, 2014----worth 10%

Final Exam---December 12, 2014----worth 40%

Assignment #1, DUE Sept 25, 2014: p.14: #7. p.30-31: #5,6,12. p.46-47: #7,8,12.

Assignment #2, DUE Oct 9, 2014: p.67-68: #4,10,11,12,13(use 12). p.82-83: #6,10,11.

Assignment #3, DUE Oct 23, 2014: p.94-96: #3,4,6,7,10,12.

Assignment #4, DUE Nov. 6, 2014: p.121-122: #2(ii),3,4,6.

Assignment #5, DUE Nov. 20, 2014: p.136-37: #1,2,3.

Chapter 2: Homomorphisms, subrings, ideals.

Chapter 3: Matrix rings, polynomial rings.

Chapter 4: Divisors, units, associates. I.D.s, U.F.D.s, P.I.D.s, Euclidean domains.

Chapter 5: Modules, homomorphisms, quotient modules, direct sums.

Chapter 6: Finitely generated modules, torsion modules, free modules.

Most of the five questions on the test will be from the material in Chapters 4, 5 and 6.

Most of the questions will be from Chapters 7 and 8.

Here is a rough guide to the 3hr exam.

1.[20] True or False.

2.[10] Given A \in M_n(R) find d_1,d_2,...,d_n etc.

3.[15] Definitions.

4.[10] Given N \subset F find F/N.

5.[8] Given A a direct sum of cyclic modules, find primary components, torsion invariants, etc.

6.[7] PID/ED question.

7.[10] examples/counter-examples.

8.[10] Linear algebra/companion matrix question.

9.[10] General module question.

**IMPORTANT SENATE POLICY:**

Students are responsible for ensuring that their selection of courses is appropriate and accurately recorded and that all course prerequisites have been successfully completed. If the student does not have the requisites for a course, and does not have written special permission from his or her Dean to enroll in the course, the student may be removed from the course and it will be deleted from the student's record. This decision may not be appealed. A student will receive no adjustment to his or her fees in the event that he or she is dropped from a course for failing to have the necessary prerequisites.

**STATEMENT ON ACADEMIC OFFENCES:**

** **"Scholastic offences are taken seriously and students are directed to
read the appropriate policy, specifically, the definition of what
constitutes a Scholastic Offence, at the following Web site:
http://www.uwo.ca/univsec/handbook/appeals/scholoff.pdf** "**

**NEW MEDICAL EXCUSE REGULATIONS:**

If you are unable to meet a course requirement due to illness or other

serious circumstances, you must provide valid medical or other

supporting documentation to the Dean's Office as soon as possible and

contact your instructor immediately. It is the student's responsibility

to make alternative arrangements with their instructor once the

accommodation has been approved and the instructor has been informed. In the event of a missed final exam, a "Recommendation of Special Examination" form must be obtained from the Dean's Office immediately.

For further information please see:

http://www.uwo.ca/univsec/handbook/appeals/medical.pdf

A student requiring academic accommodation due to illness, should use the Student Medical Certificate when visiting an off-campus medical facility or request a Record's Release Form (located in the Dean's Office) for visits to Student Health Services.

The form can be found here:

https://studentservices.uwo.ca/secure/medical_document.pdf