MATH 323 Rings and Modules.

# 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.
• 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.

### 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:
• 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).
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.

• 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.