THE UNIVERSITY OF WESTERN ONTARIO

LONDON CANADA

DEPARTMENT OF MATHEMATICS

__Mathematics 4123A/9023A: Rings and Modules__

**Fall 2015**

**
**

**INSTRUCTOR:** Lex Renner

**TEXTBOOK:** Abstract Algebra by D. S. Dummit and R. M. Foote, Wiley, 3rd Edition,
ISBN:978-0-471-43334-7.

**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:**
Rings of fractions and localization, Chinese Remainder Theorem, factorization in commutative rings,

Euclidean algorithm, PIDs, algebraic integers, polynomials and formal power series,

factorization in polynomial rings; Modules: generation, direct products and sums, freeness,

presentations, modules over PIDs. Tensor algebras, exact sequences, projectivity, injectivity,

Hom and duality, Zorn's Lemma, chain conditions.

**CHAPTERS/SECTIONS/TOPICS FROM THE TEXT:**

ESSENTIAL: Chapters 7,8,10,12, Zorn's Lemma(Appendix I), Algebraic Integers(15.3)

POSSIBLE: 9.3, 15.1, 15.5, Chapter 16.

**EVALUATION OF STUDENT PERFORMANCE:**

Assignments---worth 40%

In-class Midterm #1---October 15, 2015----worth 10%

In-class Midterm #2---November 19, 2015----worth 10%

Final Exam---December 15, 2015----worth 40%

More will be expected of 9023 students than 4123 students.

g.c.d.s, rings of fractions, prime ideals, irreducible elements, basic modules. Also definitions and counter-examples.

and primary components. Applications to linear algebra: Cayley-Hamilton, rational canonical form and Jordan canonical form.

1. True or False [20]

2. Definitions [10]

3. Direct sums [8]

4. Euclidean domain proof [8]

5. Examples and counter-examples [8]

6. Linear algebra via k[x] [8]

7. Modules [8]

8. Integral extensions proof [5]

9. Tensor products [5]

ASSUME THAT R IS A RING WITH 1 NOT EQUAL TO 0.

p.249: #13,17,19.

p.250: #29,30,34.

p.259: #36,37,38.

p.278-279: #5c),7,11.

p.282-283: #2,7,8.

p.293: #2,3,6.

p. 468 -- 469: #1, 2, 6, 7, 9, 13.

p. 488: #2, 4, 6, 7.

p. 489: #11, 14.

p. 500: # 9, 10, 12.

p. 703: #2, 5, 6, 8.

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**IMPORTANT SENATE POLICY:**

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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.

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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