MATH 323. Rings and Modules.
Text: Dummit and Foote, "Abstract Algebra".
Section 201, Instructor: Julia Gordon.
Where: BUCH A 202.
When: Tue, Th 9:30-11am.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: Tuesdays 11am-noon,
Fridays 10-11am, and by
appointment.
-
Course outline
-
The policies on marking, etc. can be found here .
- The midterm: In class on Thursday February 16 (this date is
tentative and will be confirmed during the first week of class).
Announcements
- The exam: Tuesday April 11, at noon.
- Office hours the last week of class: Monday April 3, 2-3pm, Tuesday
11-12, Thursday 11-12.
(no Friday!).
- Review session: Monday April 10, 10am - noon, in GEOG 201 (Note the
unusual room!)
Also more office hours Monday April 10, 2:30-4:30pm.
- Piazza links (sorry I have been off it for a while; coming back
now). The
sign-up
link .
The
class
link.
Review materials for the final exam
HOMEWORK
There will be weekly homework assignments, posted here every Monday, and
due the following Tuesday.
- Here are some resources if you want to
start using TeX (optional, of course).
Problem set 1 (due Tuesday January 10).
Solutions.
Problem Set 2
(due Thursday Jan. 19).
Solutions.
Problem Set 3 (due Thursday January
26).
Solutions.
Problem Set 4 (due Thursday Feb. 2).
Solutions.
Problem set 5 (due Thursday,
February 9).
Solutions .
picture for Problem 4 (with thanks to
Wikipedia for the picture).
Picture for Problem 5 .
Problem Set 6
(due Thursday March 2).
Solutions.
Problem Set 7 (due Thursday March 9)
Also recommended (but not to be
written up): problems 2,5, 6,7 from 13.1, and
problem 2 from 9.5 -- note that it refers to
problem 6 from 9.4, so it makes sense to do these two together).
Solutions.
Problem Set 8 (due March 16).
Solutions
Problem Set 9 (due March 23).
Solutions
Propblem set 10 (Due Tuesday April 4
-- note the unusual day!)
Just for fun (optional reading, not required to solve the problem) --
Problem 7 is related to an elementary Pick's Theorem .
Solutions. Picture
for
Problem 8 .
The last set of suggested problems -- you might want to look at them
before the final exam (not to be
handed
in):
- Secion 10.1, proplems 3, 12, and 15-19.
- 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.
Review materials for the midterm
The list of topics for the midterm
.
(I think the best way to use the list is look at the items with
closed
book, try to recall all the relevant definisions, facts, proofs, and
examples, and if any of this is causing difficulty, then read the relevant
section again).
midterm from 2013 .
(Ignore the last problem, we have not yet covered Gauss' Lemma fully).
Do not read the solutions, try the problems yourself! Use the solutions
only to check your work.
Solutions .
A pracice midterm . (Ignore the last
problem) Solutions
Midterm from 2014 (Ignore Problem 9).
Solutions .
Detailed Course outline
Short descriptions of each lecture and relevant additional references will be posted here as we progress.
All section numbers refer to Dummit and Foote.
- Tuesday Jan. 3 :
The basics: motivation, and
properties of the integers from a new perspective (Sections 0.2, 0.3);
Rings -- the basic definitions.
- Thursday Jan. 5 :
Basic properties of rings; zero divisors and units. Examples:
function rings, matrix rings, Hamilton's quaternions, and quadratic
integer rings. Reference:
Sections 7.1 and 7.2.
- Tuesday Jan. 10 :
One more example or rings: polynomial rings (from 7.2).
Units in 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.
- Thursday Jan. 12 :
Section 7.3: homomorphisms. The notion of an ideal.
Quotient rings.
Examples of ring homomorphisms from 7.3.
- Tuesday Jan. 17 :
The first isomorphism theorem.
Examples of quotient rings: Q[x]/(x), Q[x]/(x^2+1), Q[x]/(x^2-5)
(these were discussed in great detail); in particular, we discussed why
the resulting rings are not isomorphic to each other (for that, we
discussed a little the idea of a polnomial equation with integer
coefficients having a solution in a given ring).
Another example: reduction homomorphisms; relationship between having
solutions in Z and solutions modulo n for all integers n.
(see the discussion on p.246 in 7.3).
- Thursday Jan. 19 :
The second, third, and fourth isomorphism theorems, with examples.
Sums and products of ideals.
We finished Section 7.3!
Started 7.4 -- properties of ideals. The notion of the generating
set for an ideal. Principal ideals.
- Tuesday Jan. 24 :
Finished section 7.4. Maximal ideals; prime ideals. The
criterion for being prime/maximal. Examples (in Z[x], F[x], and Z[i]).
We discussed the quotient Z[i]/(2) and concluded that the ideal (2) was
neither prime nore maximal in Z[i]. As a side note, Z[i]/(2) is a rather
peculiar ring of 4 elements.
If you are curious about how many rings of 4 elements exist,
here is a general proof that there are 11 rings of order p^2,
for a prime p. Later we will recognize
some of these rings as quotients of polynomial rings.
This is completely optional reading
- Thursday Jan. 26 :
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. Chinese Remainder Theorem for
general rings. Started 8.1 -- defined Euclidean domains.
- Tuesday Jan. 31 :
Proof that Euclidean domains are PIDs (Principal Ideal
Domains).
Proof that Gaussian integers are a Euclidean domain, as well as the
quadratic integer ring with D=-2.
Proof that Z[\sqrt{-5}] is not a PID and
therefore, not Euclidean.
References: Section 8.1, except we skipped everything after
Theorem 4 (Please read Theorem 4 with proof; everything after it,
including Proposition 5 and the proof that
Z[1+\sqrt(-19)/2] is not Euclidean is optional), and
Section 8.2 up to (and including) Proposition 7.
We skipped everything about Dedekind-Hasse norms and the proof that
Z[1+\sqrt(-19)/2] is a PID).
Here is another reference on Euclidean rings, and this
proof:
Notes on Euclidean domains by Keith Conrad (also optional reading).
- Thursday Feb. 2 :
Unique factorization domains - Section 8.3
up to Corollary 15 on p.289.
Irreducible elements, prime elements.
Discussed in detail the examples of non-unique factorizations in some
quadratic integer rings, and how one gets non-principal ideals from these
factorizations.
- Tuesday February 7:
Finished Section 8.3 (the application to Fermat's theorem about sums of
squares), and started 9.1.
Please read Sections 9.1-9.2 (which are easy reading and I would like to
go through them quickly). Stated Gauss' Lemma (Prop. 5 on p.303). The
form stated in class was closer to Corollary 6 on p. 304 (which is in fact
an equivalent reformulation).
- Thursday February 9:
Started
talking about polynomial rings (sketched 9.1-9.2, which were
assigned as home reading).
Also, discussed Section 7.5, because we need the field of fractions for
the proof of Theorem 7 in 9.3.
- Tuesday February 14:
Started 9.3; proved Gauss' Lemma (Proposition 5 on p.303).
So far, discussed the statement and overall
strategy of the proof of Theorem 7 in 9.3 (see p. 304-305). This theorem
is the main result of this section, and it says
that R is a UFD iff R[x] is a UFD. Will finish proving this theorem next
time.
- Thursday February 16: Midterm
- Feb. 21 - 23: break.
- Tuesday Feb.28 :
Finished Section 9.3, and discussed some examples -- how polynomial
rings give
rise to field extensions. This is covered in Section 13.1.
- Thursday March 2 :
More discussion of constructing field extensions (in particular,
constructing a finite field with a given number of elements) (see 13.1,
we will cover it in lecture up to (including) Theorem 3, but it is a
good
idea to read the whole section 13.1). Please also read 9.5.
Irreducibility criteria for polynomials (Section 9.4).
Supplementary reading if you like:
A very nice explicit description of using cyclotomic polynomials to
construct finite fields appears in Section 9
here (notes by Max Naunhoffer).
A note on
cyclotomic polynomials by Paul Garrett.
See also Some extra problems of
number-theoretic flavour that can be solved using quadratic integer rings.
- Tuesday March 7 (guest lecture)
:
Modules -- Sections 10.1, 10.2. The main examples: modules over a field
are
vector spaces; modules over Z are abelian groups. The notion of a
submodule; ideals as submodules. Definition of a module homomorphism.
The notions of: submodule and quotient module were defined. Also, the
submodule criterion (Proposition 1 in 10.1) was proved.
- Thursday March 9:
Finished the discusion of the construction of finite fields started on
March
2 (see links under March 2).
Continued modules -- please read all of sections 10.1, 10.2.
Defined the notion of annihilator, which is discussed in Homework 8 (see
the homework for the definition).
Did some review of linear algebra, in particular, discussed the abstract
notion of a vector space over a field (which is a special case of a module
when the ring is a field).
- Tuesday March 14:
Finished everything in 10.1, 10.2. In particular, returned to 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 is the part
of 10.1 on pp.340-341.
Also, defined the notions of a generating set of a module, a cyclic
module, a finitely generated module. This is the first two pages of
Section 10.3.
- Thursday March 16:
Finished 10.2: discussed the four
isomorphism theorems for modules;
Defined the homomorphism group Hom(M,N) and the ring End(M).
Defined the notion of an algebra and explained why End(M) is an algebra
over the ring R (when M is an R-module).
Also discussed direct sums and products of modules.
- Tuesday March 21:
10.3 free modules;
Examples of free and non-free modules.
The notion of rank (which is defined in 12.1).
Please read all of 10.1-10.3, with
emphasis on 10.3; also please look at
Problems 20, 24 and 27 in Section
10.3.
- Thursday March 23:
Thomas discussed the proof of Theorem 4 in 12.1; Thomas' proof was closer
to the proof in the case of
Euclidean rings that is outlined in Problems 16-19 in 12.1. --
please look at these exercises and read the proof of Theorem
4, and Theorems 5-7.
- Tuesday March 28:
Will finish the discussion of free modules from last Tuesday, especially
the universal property, and will finish the discussion of when the notion
of rank is (and isn't) well-defined. -- finish Section 10.3.
Then we get back to discussing the classification of modules over PIDs
(Section 12.1)
For a lot more
information on universal properties, see
Supplement on categories by Lior Silberman.
- Thursday March 30:
Review of free modules and the statement of the main theorem for structure
of modules over PIDs. (just review of a few of the previous lectures).
- Tuesday April 4:
Discussed the example of Z^2: taking quotients by
submodules. (concrete example of the theorem about structure of modules
(this is Theorem 4 from 12.1)
Notes from this part.
Stated 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).
Invariant factors and elementary divisors.
- Thursday April 6: Quick survey of rational and Jordan canonical form.
(Sections
12.2-12.3).This is not on the exam, but this application will be also used
to review
the main points about the structure of modules over a PID.