THE UNIVERSITY OF WESTERN ONTARIO

LONDON CANADA

DEPARTMENT OF MATHEMATICS



Mathematics 4123A/9023A: Rings and Modules

Fall 2016



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.

LECTURES: TuTh 11:30 -- 1:00, MC 108. First lecture: September 8, 2016.

EVALUATION OF STUDENT PERFORMANCE:
Assignments---worth 40%
In-class Midterm #1, October 18, 2016 ---- worth 10%
In-class Midterm #2, November 22, 2016 ---- worth 10%
Final Exam, December 10, 2016, 9:00 -- 12:00am ---- worth 40%
More will be expected of 9023 students than 4123 students.

EXAMINATION DATES AND TOPICS:
Midterm 1: October 18, 2016: POSSIBLE TOPICS: Euclidean domains, principal ideal domains, unique factorization domains,
integral domains, g.c.d.s, rings of fractions, prime ideals, irreducible elements, free modules. Also definitions and counter-examples.
Midterm 2: November 22, 2016: POSSIBLE TOPICS: Finitely generated modules over PIDs: Invariant factors, elementary divisors,
and primary components. Applications to linear algebra: Cayley-Hamilton, rational canonical form and Jordan canonical form.
Final Exam:December 10, 2016, 9:00 -- 12:00am: TOPICS: Comprehensive. Here is some information about the questions from 2015. This year will be similar.
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]

ASSIGNMENTS:
Assignment #1 SCAN1 : Due: September 27, 2016 at 11:30 am..
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.
Assignment #2 SCAN2 : Due: October 13, 2016 at 11:30 am.
p.278-279: #5c),7,11.
p.282-283: #2,7,8.
p.293: #2,3,6.
Assignment #3 SCAN3 : Due November 1, 2016 at 11:30 am.
p. 468 -- 469: #1, 2, 6, 7, 9, 13.
Assignment #4 SCAN4 : Due November 17, 2016 at 11:30 am.
p. 488: #2, 4, 6, 7.
p. 489: #11, 14.
p. 500: # 9, 10, 12.
Assignment #5 SCAN5 : Due before the final exam.
p. 703: #2, 5, 6, 8.


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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.


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