Math 228 (R1)

Introduction to Ring Theory

Winter 2009

Instructor:    Matilde Lalín

Classroom:    CEB 331

Class Times:    Tuesdays - Thursdays 11:00 - 12:20 No classes on reading week: February 16-20

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

• Syllabus Office hours have been extended on Mondays to 11:30-13:30

Homework:
Unless otherwise stated, homework problems are taken from Hungerford, Abstract Algebra, Second Edition
• Homework 9, due Thursday 4/2: from section 5.1: 7, 12, from section 5.2: 11, 14b, from section 5.3: 1a,b, 9b (hint= If alpha=[x], what happens with alpha^2+alpha?), from section 6.1: 4, 18. Solutions.

Other suggested problems: from section 5.1: 6, 11, from section 5.2: 8, 14, from section 5.3: 5, 10, from section 6.1: 19, 21, 25, 42
• Homework 8, due 3/24: from section 4.5: 1d, 5b, 6, 11, 18b, from section 4.6: 1b, 6. Solutions.

Other suggested problems: from section 4.5: 1, 3, 7, 18, from section 4.6: 2, 3, 8
• Homework 7, due 3/17: from section 4.2: 3, 5f, from section 4.3: 12, 14, 23b, from section 4.4: 4a, 8f. Solutions.

Other suggested problems: from section 4.2: 4, 8, 15, from section 4.3: 15, 22, from section 4.4: 17, 18, 26
• Homework 6, due 3/10: from section 3.2: 22, from section 3.3: 9, 10 c,d,e, 19 (note that 3.1.18 was in homework 5, hint: take the function f(x)=1-x), 32 a, 33 f, from section 4.1: 3a, 10, 18. Solutions.

Other suggested problems: from section 3.2: 13, 24, 28, 30, from section 3.3: 10 a,b, 13, 18, 32 b,c, 33 a-e, from section 4.1: 6, 13, 17
• Homework 5, due 2/24: from section 3.1: 5d, 12, 18, 24, from section 3.2: 2, 5, 14. Solutions.

Other suggested problems: from section 3.1: 16, 26, 31, 35, 37, from section 3.2: 12, 19

• Homework 4, due 2/10: from section 2.1: 11b,d, from section 2.2: 8, 10, from section 2.3: 2, 7b,d,f, 11b. Solutions.

Other suggested problems: from section 2.1: 8, from section 2.2: 5, 9, from section 2.3: 5

Would you like to do more problems? Please go to Midterm preparation
• Homework 3, due 2/3: from section D: 3a, from section 2.1: 12, 16b, 19, 32, from section 13.1: 10, 17. Solutions.

Other suggested problems: sec D: 6, sec 2.1: 26, 29, sec 13.1: 5, 7, 12, 14, 18
• Homework 2, due 1/27: from section C: 2, 6, from section 1.3: 8, 12a, 14, 20, 28. Solutions.

Other suggested problems: sec C: 15, 16, sec 1.3: 7, 9, 11, 15, 22, 27
• Homework 1, due 1/20: from section 1.1: 6, 8, from section 1.2: 8, 10, 16, 26, 32. Solutions.

Other suggested problems (do not turn in): sec 1.2: 7, 14, 20, 25, 28, 34, 36

Special Announcements:
• 3/24: We'll be doing course evaluations on March 26
• 2/3: Office hours have been extended on Mondays to 11:30-13:30. I realize this is not enough for everybody, but it does help many of you.
• 1/15: Suggested problems are not to be turned in, but for you to think about besides the homework. You are welcome (in fact, encouraged) to ask me questions about them in the office hours. Also if you solved a suggested problem and you want me to read your solution, feel free to bring it to my office hours.
• 1/8: First homework is up!!!

Final Preparation:
The final will be on April 20 9:00-12:00, in the usual classroom.

The topics to be included in the final are everything that was covered up to the class of March 31 and the definition and properties of kernel (covered today, April 2). I won't ask questions about First Homomorphism Theorem, prime ideals or maximal ideals. There will be an emphasis in the topics that weren't covered in the midterm. Please consult the Topics covered in Class section for additional details on what topics were covered.

The kind of questions will be similar to the midterm questions, but there will be some true/false statements and some questions may have several items.

To practise, here are some old finals Old Final A (ignore 3c, 5a,c), Old Final B (ignore 2.2), and Old Final C and Old Final C (extra page) (the little clock in problem 3 means Z_12).

Solutions

Remember that it is always better if you think these problems without looking at the solutions. You are encouraged to come to my office hours and ask me questions about the problems in the old finals.

Office hours: it is my intention to be in my office on working days 12:30-1:30 (except on Thursday April 9). If you stop by my office at any other time and you find me there you can also ask me questions. And you can also e-mail me with questions. Please take into account that I'm VERY close to finishing my pregnancy, and therefore I may not be able to make it to office hours or answering my e-mail. The best strategy for you is to ask me questions as early as possible (i.e., there is a higher probabily to find me on April 14 than April 17, and so on). I know this is hard to plan. Rest assured that I have made plans in case someone else has to proctor the final exam. I will make every possible effort to mark it.

Midterm Preparation:
The midterm will be on March 3 in class.

The topics to be included in the midterm are the topics covered in class up to February 10, excluding 3.3 and 1.4. Roughly speaking, these are: sections 1.1-1.3, appendix D, 2.1-2.3, 13.1, 3.1. Please consult the Topics covered in Class section for additional details.

To practise, here are some old midterms Old Midterm A, Old Midterm B, (beware of the following: problem 1 has a clock to mean Z (or Z_n). problem 3: you don't know how to do this problem and it's OK. problem 2: regular means unit. problem 4: irreducible (in this context) means prime), and Old Midterm C.

Solutions to the old midterms.

It is better if you think these problems without looking at the solutions. You are encouraged to come to my office hours and ask me questions about the problems in the old midterms. Midterm Solutions.

Important dates:
• March 3: Midterm, in class
• April 20: 9:00-12:00 : Final Exam
• May 9: 9:00-12:00: Deferred Exam

Topics covered in Class:

• 4/6: practise problems for review (ask me questions!!!!!!!)
• 4/2: kernel of a homomorphism (6.2) (this is the last topic to enter in the final exam), other topics were covered for your general interest that won't be on the final exam: statement of First Homomorphism Theorem (6.2), maximal and prime ideals (6.3), A general comment about Chapter 9, which would be the natural continuation to the topics that we covered in the class. If you want to learn more about these topics you are welcome to ask me questions!
• 3/31: finitely generated ideals, congruences (6.1), quotients (6.2)
• 3/26: Course evaluations, review of the fact that p(x) has a root in F[x]/(p(x)) from 5.3, intro to ideals up to principal ideals (6.1)
• 3/24: end of 5.2, 5.3 (the structure of F[x]/(p(x)) when p(x) is irreducible)
• 3/19: properties of congruence in F[x] (5.1, 5.2, up to F is a subfield of F[x]/(p(x)))
• 3/12: Rest of 4.5 (Rational root test, products in Z versus products in Q, Eisenstein's criterion, reduction mod p)
• 3/10: Factorization into irreducibles is unique (end of 4.3), roots (remainder theorem, factor theorem, relationship between roots and irreducibility 4.4), statement of rational root test in Q (4.5).
• 3/5: units, associates, irreducibles, and factorization into irreducibles (4.3) (4.3)
• 2/26: review of Euclidean algorithm and chinese remainder theorem, and Divisibility in polynomials (4.2)
• 2/24: Polynomials and division algorithm (4.1)
• 2/12: Isomorphisms and homomorphisms, examples and properties, properties preserved by homomorphisms and isomorphisms. (3.3)

Curiosity: Speaking about vector spaces and rings (and their homomorphisms) we see that have lots in common. We can talk about all of them together in the language of Categories. We can even talk about funcions that send rings in vector spaces! (they are called functors). As you learn more algebraic structures (such as groups) you'll have a chance of studying homomorphisms and see that they "look all the same".
• 2/10: rest to 3.2 (units, zero divisors, relationship between fields and integral domains), introduction to 3.3 (isomorphisms, definition of isomorphism)

Curiosity: To know a finite field is to know its cardinality, in other (fancier) words, finite fields with the same cardinality are isomorphic.
• 2/5: Product of rings, subrings (from 3.1), basic properties of rings (3.2, additive inverse and cancelation in sums).
• 2/3: Intro to rings: definition of ring, commutative, ring with unity, integral domain, field, and many examples, including complex numbers (3.1).

Curiosities: 1) The term "ring" was introduced by Hilbert at the end of the nineteen century, but it's not clear the motivation of such choice. A little bit of history.

2) If you only leave axioms 1,2,4,5 (addition is closed, associative, with identity and every element has an inverse) and FORGET about multiplication, you're left with a GROUP. Here you may read more about them!
• 1/29: Rest of 2.3 with examples of linear equations.

• 1/27: 2.2 (Modular arithmetic, operations with Z_n), Theorem 2.8 of 2.3 (Z_p, p prime).

Curiosity: Sometimes we can say general stuff about equations in congruences of degree higher than 1. Like in Fermat's little theorem.
• 1/22: Equations in congruences, 13.1 (Chinese Remainder Theorem).
• 1/20: part of appendix D (definition of equivalence relations), rest of 2.1 (congruence classes).
• 1/15: rest of 1.3 (Fundamental Theorem of Arithmetic), a little bit of 1.4 (n is prime if it doesn't have divisors smaller or equal than sqrt(n), Sieve of Erathosthenes, Mersenne primes), introduction to 2.1 (congruences).

Curiosity: Great Internet Mersenne Prime Search.
• 1/13: part of appendix C (mathematical and complete induction, examples of mathematical induction, we stated (without proof) that woa is equivalent to induction), 1.3 (primes, and the property that if they divide a product they have to divide one of the factors).

Curiosity (just for fun): Tower of Hanoi (see also problem 16 in Appendix C of the book). You can play with it here.
• 1/8: 1.2 (divisibility, greatest common divisor, Euclidean algorithm).

Curiosity (just for fun): The Argentinian coin crisis (also here). Argentinians have become experts in computing linear combinations of integral numbers (i.e. bills) so that they don't loose coins. Here are the bills. What would happen if the 2 peso bill disappeared?
• 1/6: Introduction to syllabus and class, 1.1 (the integers, ordering in the integers, well-ordering axiom, division algorithm), intro to 1.2 (definition of divisibility).

Other Resources:

In no particular order! I'll keep updating this, so let me know of any suggestions!

Last update: April 2, 2009 (or later)