11-1100/Homework Assignment 5 - Revision history

Drorbn: /* Last Week's Schedule */

2011-12-07T14:54:54Z

Last Week's Schedule

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	\|Noon		\|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.		\|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.
			\|-
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			\|2PM
			\|Additions to this web site no longer count for good deed points after this time.
	\|-		\|-
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Tholden: /* Solve the following questions */ Spelling Mistake

2011-12-02T22:41:46Z

Solve the following questions: Spelling Mistake

← Older revision		Revision as of 18:41, 2 December 2011
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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.
	# ~~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>.		# 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>.

Drorbn at 01:16, 1 December 2011

2011-12-01T01:16:54Z

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	{{In Preparation}}
	{{11-1100/Navigation}}		{{11-1100/Navigation}}
			===The Final Exam===
			The Final Exam will take place on Friday December 9, 10-1 at Bahen 6183. You may expect very approximately one third of the exam to be about class-proven 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 [[10-1100/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===		===Last Week's Schedule===
	<center>		<center>

Drorbn at 23:24, 28 November 2011

2011-11-28T23:24:36Z

← Older revision		Revision as of 19:24, 28 November 2011
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	{{In Preparation}}		{{In Preparation}}
	{{11-1100/Navigation}}		{{11-1100/Navigation}}
	===Last Week Schedule===		===Last Week's Schedule===
	<center>		<center>
	<span style="color:red; margin:0; padding:0">		<span style="color:red; margin:0; padding:0">

Drorbn at 23:21, 28 November 2011

2011-11-28T23:21:39Z

← Older revision		Revision as of 19:21, 28 November 2011
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	===Last Week Schedule===		===Last Week Schedule===
	<center>		<center>
			<span style="color:red; margin:0; padding:0">
	'''Warning.''' This schedule is subject to changes. Recheck this web site the day before any activity.		'''Warning.''' This schedule is subject to changes. Recheck this web site the day before any activity.
			</span>

	{\|border=1 cellspacing=0		{\|border=1 cellspacing=0
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	\|Wednesday December 7		\|Wednesday December 7
	\|12-2		\|12-2
	\|{{Dror}}'s ~~Office~~ ~~Hours~~, Bahen 6178.		\|{{Dror}}'s office hours, Bahen 6178.
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	\|Thursday December 8		\|Thursday December 8
	\|10:30-12:30		\|10:30-12:30
	\|{{Dror}}'s ~~Office~~ ~~Hours~~, Bahen 6178.		\|{{Dror}}'s office hours, Bahen 6178.
	\|-		\|-
	\|		\|
	\|Noon		\|Noon
	\|HW5 due ~~date~~, to be graded after the final. Also, at this time "early bird" marked HW5 can be collected at {{Dror}}'s ~~Office~~.		\|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.
			\|-

			\|

			\|3-5
			\|Stephen Morgan's office hours, at Huron 1028.
			\|-
			\|Friday December 9
			\|10-1
			\|The Final Exam.
	\|}		\|}
			</center>

	===Solve the following questions===		===Solve the following questions===

Drorbn at 23:14, 28 November 2011

2011-11-28T23:14:30Z

New page

{{In Preparation}}
{{11-1100/Navigation}}
===Last Week Schedule===
<center>
'''Warning.''' This schedule is subject to changes. Recheck this web site the day before any activity.

{|border=1 cellspacing=0
|-
|Tuesday December 6
|10-12
|Last Class
|-
|Wednesday December 7
|12-2
|{{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
|10:30-12:30
|{{Dror}}'s Office Hours, Bahen 6178.
|-
|
|Noon
|HW5 due date, to be graded after the final. Also, at this time "early bird" marked HW5 can be collected at {{Dror}}'s Office.

|}

===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>, that <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>.