MATH:5010 - Spring 2017

Reading (from Dummit and Foote):

Part A Problems (to be handed in on Tuesday, February 21, at the beginning of the discussion section):

• Section 12.1: # 11, 12. (Note that all modules are assumed to be unital.)
  IMPORTANT for #11, 12:
  • Do NOT quote Lemma 8 or Theorem 9 on p. 466 in your proofs. These exercises give an alternative proof to the one presented on pp. 466-467. However you can use Theorem 6 on p. 464.
  • In #11: The number k is a positive integer, and n is the maximal non-negative integer such that pⁿ divides a in R. The isomorphism is an isomorphism of R-modules. One way to do this exercise is by finding an explicit R-module homomorphism and using the first isomorphism theorem when k ≤ n, and by showing that p^m M = pⁿ M for all m ≥ n.
  • In #12(a): You may use #7 WITHOUT proof for #12(a). But be sure you know how to prove #7.
  • In #12(b): Note that by definition, the elementary divisors are positive powers of primes. Hence there are no elementary divisors of the form p⁰. So the number of elementary divisors of the form p^a with a ≥ 1 is the same as the number of elementary divisors of the form p^a with a ≥ 0.

• Section 12.1: # 16.
  IMPORTANT for #16: You should write down a proof which works for an arbitrary ring R with 1 and an arbitrary unital left R-module M.

Part B Problems (to be handed in on Friday, February 24, at the beginning of the lecture):

• Section 12.1: # 15.
  IMPORTANT for #15: Assume that R is a left Noetherian ring with 1. All the modules in the hint are unital left R-modules.

• Section 12.1: # 19. (R is assumed to be a Euclidean domain and M is a unital left R-module, see the paragraph before #16.)
  IMPORTANT for #19:
  • Read the paragraph before #16 and read the paragraph between #16 and #17.
  • You may assume you know #17 and #18 WITHOUT having to prove them. Note that the statements in #17 and #18 are true for any finite set of generators for ker(ϕ).
  • Note that the elementary row and column operations referred to in part (a) of #19 are precisely those discussed in #17 and #18. But it is not allowed to just multiply a row (or a column) by an arbitrary non-zero element of R, since that operation may change the modules ker(ϕ) (resp. Rⁿ) -- that's the difference between the situation here and when you work over a field. You may multiply by a unit in R, though.
  • Let N be the degree function (or norm) on the Euclidean domain R. Part (a) is based on repeatedly using the division algorithm, which corresponds to the operations of the form #17(b) or #18(b) (why?). First use row and column switches to put a non-zero element of the matrix of minimal degree into the (1,1) entry. Now keep using the division algorithm (together with row and column switches) to find a relations matrix of the form desired in part (a). To show this process stops after finitely many steps, recall that if you use the division algorithm then the remainder is either 0 or has degree strictly smaller than the degree of the element you divide by.
  • For part (d), you may again use #7 WITHOUT proof.
  • Read the paragraph between #19 and #20. In a future exercise, you will apply the method described in # 16-19 to the special case when R = F[x] and F is a field.

Additional Practice Problems (you have to know how to do these, but you do NOT have to hand these in):

• Section 12.1: # 7, 8, 9, 13, 14, 17, 18.