Difference between revisions of "111100/Homework Assignment 5"
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'''Problem 5.''' (Dummit and Foote, page 468) Let <math>M</math> be a module over a commutative domain <math>R</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.  # 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.  
−  #  +  # Conversely 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>.  '''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>. 
Revision as of 18:41, 2 December 2011

The Final Exam
The Final Exam will take place on Friday December 9, 101 at Bahen 6183. You may expect very approximately one third of the exam to be about classproven theorems, one third to be repeats of HW problems and/or exam problems from this year or last, and one third to be fresh exercises. You will have to solve about 5 out of about 6 problems. It is likely that the overall shape of the exam will be similar to last year's final exam, which can be found at 101100/Final Exam.
When I was a student, before exams I usually made sure that I absolutely understand all class material and I worried less about the exercises, on the assumption that class was about the most important knowledge and that if I really understood all that was done in class, the exercises would follow relatively easily. That was my strategy; it worked well for me, but what works for you is not for me to tell.
Last Week's Schedule
Warning. This schedule is subject to changes. Recheck this web site the day before any activity.
Tuesday December 6  1012  Last Class 
Wednesday December 7  122  Dror's office hours, Bahen 6178. 
2PM  HW5 "early bird" due date. If you submit HW5 by this time, it will be marked by noon of the following day.  
Thursday December 8  10:3012:30  Dror's office hours, Bahen 6178. 
Noon  HW5 is due in Dror's office, to be graded after the final. Also, at this time "early bird" marked HW5 can be collected at Dror's office.  
35  Stephen Morgan's office hours, at Huron 1028.  
Friday December 9  101  The Final Exam. 
Solve the following questions
Problem 1. Let be a module over a PID . Assume that is isomorphic to , with nonzero nonunits and with . Assume also that is isomorphic to , with nonzero nonunits and with . Prove that , that , and that for each .
Problem 2. Let and be primes in a PID such that , let denote the operation of "multiplication by ", acting on any module , and let and be positive integers.
 For each of the modules , , and , determine and .
 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 modules ("Laurent polynomials in ") and (here acts on as ).
Problem 4. (from Selick) Show that if is a PID and is a multiplicative subset of then is also a PID.
Definition. The "rank" of a module over a (commutative) domain is the maximal number of linearlyindependent elements of . (Linear dependence and independence is defined as in vector spaces).
Definition. An element of a module over a commutative domain is called a "torsion element" if there is a nonzero such that . Let denote the set of all torsion elements of . (Check that is always a submodule of , but don't bother writing this up). A module is called a "torsion module" if .
Problem 5. (Dummit and Foote, page 468) Let be a module over a commutative domain .
 Suppose that has rank and that is a maximal set of linearly independent elements of . Show that is isomorphic to and that is a torsion module.
 Conversely show that if contains a submodule which is isomorphic to for some , and so that is torsion, then the rank of is .
Problem 6. (see also Dummit and Foote, page 469) Show that the ideal in , regarded as a module over , is finitely generated but cannot be written in the form .