14-1100/Homework Assignment 5

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This assignment is extended from class time on Wednesday, December 3, 2014 (a "virtual Monday" and the last day of the semester) to the end of Monday, December 8 in Dror's mailbox.

Solve the following questions

Problem 1. Let M be a module over a PID R. Assume that M is isomorphic to R^k\oplus R/\langle a_1\rangle\oplus R/\langle a_2\rangle\oplus\cdots\oplus R/\langle a_l\rangle, with a_i non-zero non-units and with a_1\mid a_2\mid\cdots\mid a_l. Assume also that M is isomorphic to R^m\oplus R/\langle b_1\rangle\oplus R/\langle b_2\rangle\oplus\cdots\oplus R/\langle b_n\rangle, with b_i non-zero non-units and with b_1\mid b_2\mid\cdots\mid b_l. Prove that k=m, that l=n, and that a_i\sim b_i for each i.

Problem 2. Let q and p be primes in a PID R such that p\not\sim q, let \hat{p} denote the operation of "multiplication by p", acting on any R-module M, and let s and t be positive integers.

  1. For each of the R-modules R, R/\langle q^t\rangle, and R/\langle p^t\rangle, determine \ker\hat{p}^s and (R/\langle p\rangle)\otimes\ker\hat{p}^s.
  2. Explain why this approach for proving the uniqueness in the structure theorem for finitely generated modules fails.

Problem 3. (comprehensive exam, 2009) Find the tensor product of the {\mathbb C}[t] modules {\mathbb C}[t,t^{-1}] ("Laurent polynomials in t") and {\mathbb C} (here t acts on {\mathbb C} as 0).

Problem 4. (from Selick) Show that if R is a PID and S is a multiplicative subset of R then S^{-1}R is also a PID.

Definition. The "rank" of a module M over a (commutative) domain R is the maximal number of R-linearly-independent elements of M. (Linear dependence and independence is defined as in vector spaces).

Definition. An element m of a module M over a commutative domain R is called a "torsion element" if there is a non-zero r\in R such that rm=0. Let \mbox{Tor }M denote the set of all torsion elements of M. (Check that \mbox{Tor }M is always a submodule of M, but don't bother writing this up). A module M is called a "torsion module" if M=\mbox{Tor }M.

Problem 5. (Dummit and Foote, page 468) Let M be a module over a commutative domain R.

  1. Suppose that M has rank n and that x_1,\ldots x_n is a maximal set of linearly independent elements of M. Show that \langle x_1,\ldots x_n\rangle is isomorphic to R^n and that M/\langle x_1,\ldots x_n\rangle is a torsion module.
  2. Conversely show that if M contains a submodule N which is isomorphic to R^n for some n, and so that M/N is torsion, then the rank of M is n.

Problem 6. (see also Dummit and Foote, page 469) Show that the ideal \langle 2,x\rangle in R={\mathbb Z}[x], regarded as a module over R, is finitely generated but cannot be written in the form R^k\oplus\bigoplus R/\langle p_i^{s_i}\rangle.