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.

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HOMEWORK

There will be weekly homework assignments, posted here every Monday, and due the following Tuesday.
  • Problem set 1 (due Tuesday January 10). Solutions.
  • Problem Set 2 (due Thursday Jan. 19).
  • Problem Set 3 (due Thursday January 26).

    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 : 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".