Math 228 (R1)

Introduction to Ring Theory

Winter 2010

Instructor:    Matilde Lalín

Classroom:    CEB 331

Class Times:    Tuesdays - Thursdays 11:00 - 12:20 No classes on reading week: February 15-19

Office:    CAB 621

Office hours:   Mondays 11:30 - 13:30, Tuesdays 10:00 - 11:00, Thursdays 12:30 - 13:30, and by appointment.

Phone:   (780) 492-3613

e-mail:    lalin at ualberta . ca or mlalin at math . ualberta . ca

Text:    T. W. Hungeford, Abstract Algebra: An Introduction

Important links:

Unless otherwise stated, homework problems are taken from Hungerford, Abstract Algebra, Second Edition

Final Preparation:

Midterm Preparation:

Special Announcements:
  • 3/25: I have added additional office hours on Wednesday March 31 from 11:30-13:30, because the last homework is due on Thursday April 1.
  • To answer some concerns from the survey: the problems in the exams will be of similar difficulty as the ones in the homework. I will not ask you to re prove things that were done in class.
  • Office hours have been extended on Mondays to 11:30-1:30 to accomodate everybody who has filled the survey. In addition, I can give office hours on Tuesdays 1:30-2 by appointment (that means, I will make sure I'm in my office on Tuesdays 1:30-2 if you let me know in advance that you're coming. You can let me know by sending me an e-mail, or talking to me in class, or calling my office).
    • I will not accept late homework. There are no exceptions to this. If you know you will be out of town when the homework is due, you have a couple of options: 1) you can turn the homework earlier or 2) you can either scan the homework and send it to me by e-mail or you can fax it to me. As long as I receive it before 12:20 the day it is due, I will accept it.

    Important dates:
    • March 2: Midterm, in class
    • April 23: 9:00-12:00 : Final Exam
    • May 8: 9:00-12:00 : Deferred Exam

    Topics covered in Class:

    • 4/8: Review
    • 4/1: 5.3: the case where p(x) is irreducible, second half of 6.2 (homomorphisms, statement of the First Isomorphism Theorem)
    • 3/30: Course evaluations, first half of 6.2: R/I is a ring, 5.2 F[x]/(p(x)) is a ring containing F
    • 3/25: continuation of 6.1: properties of congruences mod I (it is an equivalence relation, we can sum and multiply side by side, the set of representatives of the cosets (called S in class), with special emphasis for the representatives of the cosets in the polynomials as in 5.1), examples of R/I for polynomials with coefficients in a finite field (we get that R/I is finite and we can count the number of elements) and R[x]/(x^2+1), which is isomorphic to C (but we didn't prove that)
    • 3/23: Continuation of 6.1: criterion to prove that something is an ideal, finitely generated/principal ideals, Euclidean domains and principal ideal domains (definition, discussion that Z is principal ideal domain because it is euclidean, comment that F[x] is also pid and ed, and that Z[x] is neither) congruences mod I (keeping in mind the example of F[x], from chapter 5)
    • 3/18 rest of 4.6: irreducibles and roots in R[x], intro to 6.1 Ideals: definitions and examples.
    • 3/16: Rest of 4.5 (products in Z versus products in Q, Eisenstein's criterion, reduction mod p), 4.6 irreducibles and roots in C, a complex number is a root of a real polynomial iff its conjugate is also a root.
    • 3/11: roots (remainder theorem, factor theorem, relationship between roots and irreducibility 4.4), rational root test in Q (4.5).
    • 3/9: units, associates, irreducibles, and factorization into irreducibles (4.3), polynomials as functions, definition of root (4.4)
    • 3/4: returned midterm, existence of the division algorithm for polynomials and Divisibility in polynomials, comments of Euclidean algorithm, greatest common divisor (4.2), units in R[x] for R an integral domain (beginning of 4.3)
    • 2/25 Review for the midterm including: regular and complete induction, Chinese Remainder Theorem, Equivalence relations, general comments about uniqueness and existence
    • 2/23 Continuation of proving that two rings are not isomorphic, Polynomials and uniqueness of the division algorithm (4.1)
    • 2/11: Most of 3.3 (isomorphisms, homomorphisms, definition of isomorphism, properties that are preserved by homomorphisms), introduction to properties that are preserved by isomorphisms and proving that two rings are not isomorphic.
    • 2/9: Basic properties of rings (3.2, additive inverse and cancelation in sums, units, zero divisors, relationship between fields and integral domains),
    • 2/4: More examples, Product of rings, subrings (from 3.1),
    • 2/2: Complex numbers, 3.1 introduction to rings: definition, commutative, ring with unity, integral domain, field, examples.

      This class will be taught by Kaneenika Sinha.
    • 1/28: back to linear equations with congruences, examples. We ended up with section 2.3 and we discussed problems 9 and 10 from sectiuon 2.3
    • 1/26: 2.2 (Modular arithmetic, operations with Z_n), 2.3 (Z_p, p prime),
    • 1/21: 13.1 Chinese Remainder Theorem, equations like x^2 conguent to something.
    • 1/19: 2.1 Congruences and congruence classes.
    • 1/14: rest of 1.3 (Fundamental Theorem of Arithmetic), part of appendix D (definition of equivalence relations).

      This class will be taught by Paul Buckingham as I am here.
    • 1/12: the greatest common divisor as linear combination with integral coefficients, part of appendix C (mathematical and complete induction, examples of mathematical induction, 1.3 (primes and the property that if they divide a product they have to divide one of the factors).
    • 1/7: continuation of division algorithm, 1.2 (divisibility, greatest common divisor, Euclidean algorithm).
    • 1/5: introduction to syllabus and class, goals of the class: rings and mathematical proofs (discussion about theorem, proposition, lemma, corollary, direct proof and proof by contradiction, see appendix A for more details), 1.1 (the integers, ordering in the integers, well-ordering axiom, division algorithm).

    Other Resources:

    In no particular order! I'll keep updating this, so let me know of any suggestions!
    • Another interesting book in the topic:

      Irving R.; Integers, Polynomials, and Rings.. A Course in Algebra (Springer, 2004)

    Last update: March 18, 2010 (or later)