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
Important links:
- 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
Please go to Midterm preparation
- 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.
Curiosity: Congruences are crutial for Public-Key Cryptography. You can
learn more about this in Chapter 12 of the book.
- 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)