Introduction to Ring Theory
Classroom: CEB 331
Class Times: Tuesdays - Thursdays 11:00 - 12:20
No classes on reading week: February 15-19
Mondays 11:30 - 13:30, Tuesdays 10:00 - 11:00, Thursdays
12:30 - 13:30, and by appointment.
lalin at ualberta . ca or mlalin at math . ualberta . ca
Text: T. W. Hungeford, Abstract Algebra: An
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.
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
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,
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,
- 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.
Other suggested problems: sec 2.1: 8, sec 2.2: 5, 6, 9, sec 2.3: 5
- Homework 3,
from section D: 3a, from section 2.1: 12,
16b, 19, from section 13.1: 8, 10, 17.
Other suggested problems:
sec D: 6, sec 2.1: 26, 29, 32, sec 13.1: 5, 7,
12, 14, 18
- Homework 2,
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
for any integer n> 0.
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
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
To practise, here are some old finals Last year's final, Old Final A (ignore 3c,
Old Final B (ignore 2.2),
Old Final C and Old Final C (extra
little clock in problem 3 means Z_12).
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
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,
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),
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
midterm and solutions to
the other old midterms
- 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
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.
- March 2: Midterm, in class
- April 23: 9:00-12:00 : Final Exam
- May 8: 9:00-12:00 : Deferred Exam
Topics covered in
- 4/6: First Isomorphism Theorem, examples This
class will be
- 4/1: 5.3: the case where p(x)
is irreducible, second half of 6.2 (homomorphisms, statement of the First
- 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
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,
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.
rest of 1.3 (Fundamental Theorem of Arithmetic),
part of appendix D (definition of equivalence relations).
This class will be taught by
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
- 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).
In no particular order! I'll keep updating this, so let me know of any
- Another interesting book in the topic:
Irving R.; Integers, Polynomials, and Rings.. A Course in Algebra (Springer, 2004)
Last update: March 18, 2010