MATH 323. Rings and Modules.
Text: Dummit and Foote, "Abstract Algebra".
Section 101, Instructor: Julia Gordon.
Where: Chemistry building, Room D300.
When: Tue, Th 2-3:30pm.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: Tuesday 3:30-4:30pm, and by
Office hours the week of the exam: Tuesday April 9, 10:30-11:30,
Wednesday April 10, 4-5:30 (sorry, the morning office hour cancelled).
Review materials for the final exam
There will be weekly homework assignments, posted here every Wednesday, and due the following Thursday.
Problem set 1 (due Thursday January 10):
- Here are some resources if you want to
start using TeX (optional, of course).
- Thanks to a volunteer for providing the typed-up solutions to
Problem Set 2 (due Thursday Jan. 17).
Problem Set 3 (due Thursday Jan. 24).
Problem Set 4 (due Thursday Jan. 31).
Problem set 5:
- Section 0.1 (p.4), Problem 6
- Section 0.2 (pp. 7-8), Problems 1(c), 10
- Section 0.3 (pp.11-12), Problems 3, 5, 8, 14, 15(a),
- Section 7.1 (pp.230-231), Problems 13, 15.
- Solutions .
Problem Set 6 (due Tuesday Feb.
Problem Set 7 (Due Tuesday March 5):
- Section 8.1: problems 7,9, 11.
- Section 8.2: problems 1,3,5.
- Section 8.3: problem 8.
Problem Set 8 (Due Thursday March 14):
- Section 13.1: Problems 1, 3, 8. Also recommended (but not to be
written up): problems 2,5, 6,7 from this section.
- Section 9.4: Problems 1, 2(except b), 6, 9, 12;
- Section 9.5: Problems 3, 5, 6. (Also recommended, but not to be
written up, is problem 2 -- note that it refers to problem 6 from 9.4,
so it makes sense to do these two together).
- Solutions . For the discussion of the
last problem, please see
this post at
Keith Conrad's page.
Problem set 9 (due Thursday March 22).
Propblem set 10, part 1 (due Tuesday
2). Part 2 (about modules over polynomial rings, and matrices) will be
posted later this week, after we cover some of this material in lecture,
and will be not for handing in.
The last set of suggested problems (the "Part 2", not to be handed
- Section 10.1: Problems 8,9,10,11,13, 20. (also recommended in this
section, but not to be handed in, proplems 3, 12, and 15-19).
- Section 10.2: Problems 4,6,8,9.
- Section 12.2: 1,2,3,4,6,8,9,10,18.
- Section 12.3: 1,2,4,5,9,12,17,19,20,22.
Detailed Course outline
Short descriptions of each lecture and relevant references will be posted here as we progress.
- Thursday Jan. 3 :
The basics: properties of the integers (Sections 0.2, 0.3);
Rings -- the basic definitions and examples. (some of Section 7.1)
Hamilton's quaternions. Quadratic integer rings (if time permits).
- Tuesday Jan. 8 :
Basic properties of rings; zero divisors and units. Examples:
function rings, and quadratic integer rings. Reference: Section 7.1
(will finish the discussion of quadratic integer rings next class).
- Thursday Jan. 10 :
Unites in the quadratic integer rings. Here is a
completely optional, and not to be handed in, problem set exploring
solutions to Pell's equation (using continued
fractions), which is equivalent to finding the units in the corresponding
quadratic integer ring.
Then we introduced other examples: matrix rings, polynomial rings and
group rings (all of Section 7.2).
Started 7.3 -- defined homomorphisms and isomorphisms.
- Tuesday Jan. 15 :
Section 7.3 continued: homomorphisms; the first isomorphism theorem.
We discussed all the examples of ring homomorphisms from 7.3.
- Thursday Jan. 17 :
Finish Section 7.3 -- sums and products of ideals.
Start 7.4 -- properties of ideals. Principal ideals.
- Tuesday Jan. 22 :
Section 7.4, continued. Maximal ideals; prime ideals. The
criterion for being prime/maximal. Examples (in Z[x], F[x], and Z[i]).
Started Section 7.6 "Chinese remainder Theorem". -- did an example of
a problem that amounts to Chinese Remainder Theorem for Z, with careful
analysis of what the solution requires.
A question of "how many finite rings of order 4 exist" came up -- one of
such rings appeared as a quotient of Z[i].
Here is a general proof that there are 11 rings of order p^2,
for a prime p. (Shamil, thank you for this interesting link). Later we will recognize
some of these rings as quotients of polynomial rings.
- Thursday Jan. 24 :
7.6 -- Chinese remainder Theorem for general rings.
Start 8.1 - Euclidean domains.
- Tuesday Jan. 29 :
Proved that Gaussian integers are a Euclidean domain, as well as the
quadratic integer ring with D=-2. Proved that a Eucliden domain is a PID.
Proved that the quadratic integer ring with D=-5 is not a PID and
therefore, not Euclidean. Started on the definitions from 8.3 --
irreducible elements, prime elements. Sections: 8.1, 8.2, 8.3, but we have
not covered everything in them yet, and will be returning to 8.2 after
- Jan. 31 -- Feb. 7:
ideal domains and unique factorization
domains, continued. There was no class February 5.
Covered all in sections 8.2 -8.3.
Notes on Euclidean domains by Keith Conrad were also used.
- February 12: :
Started talking about polynomial rings (sketched 9.1-9.2, which were
assigned as home reading, and started 9.3). To prove the main result --
that R is a UFD iff R[x] is a UFD, the notion of field of fractions was
needed. So, half the lecture was spent on Section 7.4.
- Thursday Feb.14 :
Finished Section 9.3, and discussed examples -- how polynomial rings give
rise to field extensions. This is covered in Section 13.1.
- Feb. 19-21: break.
- Tuesday Feb.26 :
More discussion of constructing field extensions (in particular,
constructing a finite field with a given number of elements) (see 13.1),
irreducibility criteria for polynomials (Section 9.4). Please also read
Supplementary reading if you like: A note on
cyclotomic polynomials by Paul Garrett.
- Feb. 28: Midterm .
List of topics to review for the midterm.
- Tuesday March 5 and Wed Mar 6
Modules -- Section 10.1. The main examples: modules over a field are
vector spaces; modules over Z are abelian groups. The notion of a
submodule; ideals as submodules. Started discussion modules over the ring
F[x] (these are a vector space with a linear operator acting on it).
- March 12 -- March 13:
Continued the discussion of modules over F[x] (namely, that a module over
F[x] is a vector space with a linear operator on it) -- this was the part
of 10.1 on pp.340-341, expanded with examples. We also proved that two
such modules (V, T_1) and (V, T_2) are isomorphic if and only if there are
two bases of V such that the matrices of T_1 and T_2 with respect to these
bases are the same. Finsihed 10.1 and 10.2 -- homomorphisms of modules,
quotient modules. Mandatory home reading (from 10.2) -- the four
isomorphism theorems for modules.
- Thursday March 14:
10.3 -- generation of modules; free modules. We basically covered exactly
Section 10.3, and Proposition 3 from 12.1. Will do some more examples next
- Tuesday March 19:
More about free modules, and universal properties in general. Examples of
free and non-free modules.
Infinite direct sums and products.
The notion of rank (which is defined in 12.1).
please see Problems 20, 24 and 27 in Section 10.3.
- Thursday March 21:
Chinese Remainder Theorem for modules. Decomposition into p-primary
components. In the text, this roughly corresponds to Problems 16-18 from
10.3, and Theorem 7 from 12.1 (and some text around it).
- Tuesday March 26:
Structure of Modules over PIDs.
Theorem 4 from 12.1; examples (please see Problems 16-19 in 12.1. We will
discuss this approach first, and then prove the general theorem about
structure of modules over PIDs in all its forms (Theorems 5-7 from 12.1).
- Thursday March 28:
Finished the discussion of 12.1; discussed how to compute the quotient
of a free module by a submodule.
Started rational canonical form (Section 12.2).
- April 2-4 : Rational and Jordan canonical form. (Sections 12.2-12.3)