MATH 323. Rings and Modules.
Text: Dummit and Foote, "Abstract Algebra".
Section 201, Instructor: Julia Gordon.
Where: MATH 103 (please note the room change).
When: Tue, Th 9:30-11am.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: by
The policies on marking, etc. can be found here .
- The midterm: In class on February 27.
- The final exam will be on April 12, 3:30-6pm, in MATH 105.
If this creates a hardship (even if it does not formally qualify) please
talk to me about alternative options.
The alternate exam will be on April 30, time and location to be
determined. If you would like to take the alternate exam, please e-mail me
(unless you already have).
Review materials for the final exam
There will be weekly homework assignments, posted here every Wednesday, and due the following Thursday.
Acknowledgement: Thanks to Shamil Asgarli for writing up
solutions to most of the homework problems that will be posted here.
Problem set 1 (due Thursday January 16).
Problem Set 2
(due Thursday Jan. 23).
Problem Set 3 (due Tuesday Feb. 4).
Problem Set 4 (due Thursday Feb. 6).
Problem set 5 (due Thursday, February 14):
Section 8.1: problem 10; Section 8.2: problems 1,3,5; Section 8.3: problem
Problem Set 6 (due Tuesday
Section 9.1: Problems 4,6,13, 18;
Section 9.2: Problems 1,3,5,8.
Problem Set 7 (Due Tuesday March 11) -- see the list below:
- Here are some resources if you want to
start using TeX (optional, of course).
Problem Set 8
(Due Tuesday March 18)
- Section 13.1: Problems 1, 3. 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(a,b,c only), 9;
- Section 9.5: Problems 3. (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).
Also recommended but not to be handed in: Secion 10.1, proplems 3, 12,
Problem set 9
(due Thuesday April 1).
Propblem set 10
If you want to turn it in, please do so by Tuesday
This problem set is optional -- if you turn in some of the problems, you
get extra credit towards homework score, but you do not have to turn it
Note also a related an elementary Pick's Theorem .
Solutions. Picture for
Problem 3 .
This is just a loose collection of extra porblems (some hard, some easy)
that can be used for review and general curiosity; also some links to
things related to, but not covered in the course.
Please do not hand in any of these problems.
Many problems and links that will appear in this section were communicated
to me by Shamil Asgarli.
Some number-theoretic extra problems.
For the discussion of Problem 4, please see
this post at
Keith Conrad's page (see also the discussion of cyclotomic polynomials in
For notes on Cyclotomic polynomials, see A note on
cyclotomic polynomials by Paul Garrett.
Some extra problems on Chapter 7.
Solutions will be posted here during the break.
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.
Notes on Euclidean domains by Keith Conrad (also optional reading).
Contains the proof of the example of a PID that is not Euclidean.
The optional problem set on Pell's equation
on units in quadratic integer rings with D>0).
Review materials for the midterm (still relevant for the final)
The list of topics for the midterm .
(I think the best way to use the list is look at the items with
book, try to recall all the relevant definisions, facts, proofs, and
examples, and if any of this is causing difficulty, then read the relevant
Last year's midterm .
Do not read the solutions, try the problems yourself! Use the solutions
only to check your work.
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. 7 :
The basics: properties of the integers (Sections 0.2, 0.3);
Rings -- the basic definitions and examples. (some of Section 7.1)
Matrix rings; Hamilton's quaternions.
- Thursday Jan. 9 :
Basic properties of rings; zero divisors and units. Examples:
function rings, group rings, and quadratic integer rings. Reference:
Sections 7.1 and 7.2.
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.
Started 7.3 -- defined homomorphisms and isomorphisms.
- Tuesday Jan. 14 :
One more example or rings: polynomial rings (from 7.2).
Section 7.3 continued: homomorphisms. The notion of an ideal. Quotient
rings. The first isomorphism theorem.
Examples of ring homomorphisms from 7.3.
- Thursday Jan. 16 :
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).
The second, third, and fourth isomorphism theorems, with examples.
Thus, we finished Section 7.3, except for sums and products of ideals.
- Tuesday Jan. 21 :
Finish Section 7.3 -- sums and products of ideals.
Start 7.4 -- properties of ideals. The notion of the generating
set for an ideal. Principal ideals.
- Thursday Jan. 23 :
Section 7.4, continued. Maximal ideals; prime ideals. The
criterion for being prime/maximal. Examples (in Z[x], F[x], and Z[i]).
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
- Jan. 28-30: NO CLASS.
- Tuesday Feb. 4 :
Euclidean domains, and PIDs (Principal Ideal Domains).
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.
Reference: Section 8.1, except we have skipped everythign starting
from Theorem 4. Please read Theorem 4; Proposition 5 and the proof that
Z[1+\sqrt(-19)/2] is not Euclidean is optional.
- Thursday Feb. 6 :
Section 8.2 -- Principal Ideal Domains; we stopped at Proposition 8. The
rest of this section starting from Prop. 9 is optional (it proves that
the ring 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).
Started on the definitions from 8.3 --
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
Stopped approximately at Proposition 11 on p.284.
- February 11: NO CLASS.
Please read Sections 9.1 and 9.2, and 7.5
- February 13: :
Section 8.3 up to the end of Theorem 14 (on p. 289); omitted the rest of
this section, for now. Started
talking about polynomial rings (sketched 9.1-9.2, which were
assigned as home reading, and started 9.3, stopping at the statement of
Gauss' Lemma (Proposition 5 on p.303). Proof of it -- next time.
Also, discussed Section 7.5, because we need the field of fractions for
the proof of Theorem 7 in 9.3. 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.
- Feb. 19-21: break.
- Tuesday Feb.25 :
Finished Section 9.3, and discussed examples -- how polynomial
rise to field extensions. This is covered in Section 13.1.
Irreducibility criteria for polynomials (Section 9.4).
- Thursday Feb.27 :
List of topics to review for the midterm.
- Tuesday March 4 :
More discussion of constructing field extensions (in particular,
constructing a finite field with a given number of elements) (see 13.1,
we covered 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.
One more application/review: representing primes as sums of two squares
(See the end of Section 8.3, starting p.289).
Supplementary reading if you like:
A note on
cyclotomic polynomials by Paul Garrett.
A very nice explicit description of using cyclotomic polynomials to
construct finite fields appears
here (notes by Max Naunhoffer).
See also Some extra problems of
number-theoretic flavour that can be solved using quadratic integer rings.
- Thursday March 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).
- Tuesday March 11:
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 discussed (and left
the formal proof for the homework) 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.
- Thursday March 13:
10.2 -- homomorphisms of modules, submodules, quotient modules. Mandatory
home reading (from 10.2) -- the four
isomorphism theorems for modules.
- Tuesday March 18-20:
10.3 -- generation of modules; free modules;
the universal property of
free modules. For a lot more
information on universal properties, see
Supplement on categories by Lior Silberman.
- March 25:
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.
- March 27:
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).
Started discussing structure of Modules over PIDs (12.1).
Started discussing Theorem 4 from 12.1; example
-- submodules of Z^2.
- Tuesday April 1 (seriously):
Continued discussiing the example of Z^2: taking quotients by
submodules. Discussed the proof of Theorem 4 in 12.1 in the case of
Euclidean rings that is outlined in Problems 16-19 in 12.1.
Also quickly went over the general proof (please read the proof of Theorem
4, and Theorems 5-7).
- Thursday April 3:
Will finish the discussion of 12.1; invariant factors and elementary
- Tuesday April 8: Quick survey of rational and Jordan canonical form.