View source for 10-1100/Homework Assignment 5

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This assignment is due by the end of my office hours (at 12:30) on Thursday December 9, 2010.

===Solve the following questions===

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

'''Problem 2.''' Let <math>q</math> and <math>p</math> be primes in a PID <math>R</math> such that <math>p\not\sim q</math>, let <math>\hat{p}</math> denote the operation of "multiplication by <math>p</math>", acting on any <math>R</math>-module <math>M</math>, and let <math>s</math> and <math>t</math> be positive integers.
# For each of the <math>R</math>-modules <math>R</math>, <math>R/\langle q^t\rangle</math>, and <math>R/\langle p^t\rangle</math>, determine <math>\ker\hat{p}^s</math> and <math>(R/\langle p\rangle)\otimes\ker\hat{p}^s</math>.
# 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 <math>{\mathbb C}[t]</math> modules <math>{\mathbb C}[t,t^{-1}]</math> ("Laurent polynomials in <math>t</math>") and <math>{\mathbb C}</math> (here <math>t</math> acts on <math>{\mathbb C}</math> as <math>0</math>).

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

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

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

'''Problem 5.''' (Dummit and Foote, page 468) Let <math>M</math> be a module over a commutative domain <math>R</math>.
# Suppose that <math>M</math> has rank <math>n</math> and that <math>x_1,\ldots x_n</math> is a maximal set of linearly independent elements of <math>M</math>. Show that <math>\langle x_1,\ldots x_n\rangle</math> is isomorphic to <math>R^n</math> and that <math>M/\langle x_1,\ldots x_n\rangle</math> is a torsion module.
# Converesely show that if <math>M</math> contains a submodule <math>N</math> which is isomorphic to <math>R^n</math> for some <math>n</math>, and so that <math>M/N</math> is torsion, then the rank of <math>M</math> is <math>n</math>.

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