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

Homework:
Unless otherwise stated, homework problems are taken from Hungerford, Abstract Algebra, Second Edition
• Homework 9 due THURSDAY 4/01: from section 6.1: 3, 4, 18, from section 5.1: 7, 12, from section 5.2: 11, 14b. Solutions

Other suggested probems: from section 5.1: 6, 11, from section 5.2: 8, from section 5.3: 1a,b, 5, 9b, 10, from section 6.1: 19, 21, 25, 42
• Homework 8, due 3/23: 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 3/16: from section 4.2: 3, 5d, 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 3/9: 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 2/23: from section 3.1: 5d, 18, 24, from section 3.2: 2, 5, 14, 22. Solutions

Other suggested problems: from sec 3.1: 12, 16, 26, 31, 35, 37, from sec 3.2: 12, 19
• Homework 4, due 2/9: 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: sec 2.1: 8, sec 2.2: 5, 6, 9, sec 2.3: 5
• Homework 3, due 2/2: from section D: 3a, from section 2.1: 12, 16b, 19, from section 13.1: 8, 10, 17. Solutions

Other suggested problems: sec D: 6, sec 2.1: 26, 29, 32, sec 13.1: 5, 7, 12, 14, 18
• Homework 2, due 1/26: from appendix C: 2, 6, from section 1.3: 8, 12a, 14, 20. In addition: Using induction, prove that the sum of the first n odd positive numbers is n^2. In other words, prove that 1+3+...+(2n-1)=n^2, for any integer n> 0. Solutions

Other suggested problems (do not turn in): sec C: 15, 16, sec 1.3: 7, 9, 11, 15, 22, 27, 28
• Homework 1, due 1/19: from section 1.1: 6, 8, from section 1.2: 7, 8, 10, 16, 26. Solutions

Other suggested problems (do not turn in): sec 1.2: 14, 20, 25, 32, 34

Final Preparation:
The final will be on April 23 9:00-12:00, in the usual classroom.

Office hours before the final exam: Monday, Wendesday and Thursdays 11:30-1:30. If you see my office open at any other time, you're welcome to come in.

The topics to be included are what was covered in class until April 1, including homomorphisms but excluding the First Isomorphism Theorem. For more details, see the Topics covered in class session.

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 Last year's final, 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.
Please note that au+bv=d and (u,v)=1 does NOT imply that d=(a,b). For example, take a=6, b=4, u=2, v=-1, then au+bv=12-4=8 but (6,4)=2

Midterm Preparation:
The midterm will be on March 2 in class.

The topics to be included in the midterm are the topics covered in class up to (and including) the class of February 11 including isomorphisms and properties that are preserved by homomorphisms, but I will not ask you to prove that two rings are not isomorphic. Please consult the Topics covered in Class section for additional details.

To practise, here are some old midterms: Last year's midterm, 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.

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.

Solution to last year's midterm and solutions to the other old midterms

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)