10-1100/Homework Assignment 4 - Revision history

Drorbn at 14:01, 25 November 2010

2010-11-25T14:01:55Z

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	'''Problem 5.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements.		'''Problem 5.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements.

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⚫	'''Problem 6.''' Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.

		⚫	'''Problem 6.''' (Withdrawn, do not submit) Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.

Drorbn at 13:30, 17 November 2010

2010-11-17T13:30:45Z

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	{{10-1100/Navigation}}		{{10-1100/Navigation}}
	{{In Preparation}}

	This assignment is due at class time on Tuesday, November 30, 2010.		This assignment is due at class time on Tuesday, November 30, 2010.

Drorbn at 11:30, 17 November 2010

2010-11-17T11:30:44Z

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	'''Problem 6.''' Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.		'''Problem 6.''' Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.

	'''Problem 7.''' (Dummit and Foote) Prove that the quotient of a PID by a prime ideal is again a PID.

Drorbn at 02:28, 17 November 2010

2010-11-17T02:28:55Z

← Older revision		Revision as of 22:28, 16 November 2010
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	'''Problem 1.''' Prove that a ring <math>R</math> is a PID iff it is a UFD in which <math>\gcd(a,b)\in\langle a, b\rangle</math> for every non-zero <math>a,b\in R</math>.		'''Problem 1.''' Prove that a ring <math>R</math> is a PID iff it is a UFD in which <math>\gcd(a,b)\in\langle a, b\rangle</math> for every non-zero <math>a,b\in R</math>.

			'''Problem 2.''' (Selick) In a ring <math>R</math>, and element <math>x</math> is called "nilpotent" if for some positive <math>n</math>, <math>x^n=0</math>. Let <math>\eta(R)</math> be the set of all nilpotent elements of <math>R</math>.
			# Prove that if <math>R</math> is commutative then <math>\eta(R)</math> is an ideal.
			# Give an example of a non-commutative ring <math>R</math> in which <math>\eta(R)</math> is not an ideal.

			'''Problem 3.''' (comprehensive exam, 2009) Let <math>A</math> be a commutative ring. Show that a polynomial <math>f\in A[x]</math> is invertible in <math>A[x]</math> iff its constant term is invertible in <math>A</math> and the rest of its coefficients are nilpotent.

			'''Problem 4.''' (Lang) Show that the ring <math>{\mathbb Z}[i]=\{a+ib\colon a,b\in{\mathbb Z}\}\subset{\mathbb C}</math> is a PID and hence a UFD. What are the units of that ring?

			'''Problem 5.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements.

			'''Problem 6.''' Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.

			'''Problem 7.''' (Dummit and Foote) Prove that the quotient of a PID by a prime ideal is again a PID.

Drorbn at 14:07, 16 November 2010

2010-11-16T14:07:04Z

← Older revision		Revision as of 10:07, 16 November 2010
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	===Solve the following questions===		===Solve the following questions===

			'''Problem 1.''' Prove that a ring <math>R</math> is a PID iff it is a UFD in which <math>\gcd(a,b)\in\langle a, b\rangle</math> for every non-zero <math>a,b\in R</math>.
	'''Problem 1.'''

Ecliff at 21:44, 15 November 2010

2010-11-15T21:44:39Z

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	'''Problem 1.'''		'''Problem 1.'''

	* Dear Prof. Bar-Natan, what is Problem 1?
	** Dear Dror, just wait and see. At the moment I have no clue, but it will have some math: <math>\phi:G\to H</math>. [[User:Drorbn\|Drorbn]] 16:25, 15 November 2010 (EST)

Drorbn at 21:27, 15 November 2010

2010-11-15T21:27:02Z

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	* Dear Prof. Bar-Natan, what is Problem 1?		* Dear Prof. Bar-Natan, what is Problem 1?
	** Dear Dror, just wait and see. At the moment I have no clue. [[User:Drorbn\|Drorbn]] 16:25, 15 November 2010 (EST)		** Dear Dror, just wait and see. At the moment I have no clue, but it will have some math: <math>\phi:G\to H</math>. [[User:Drorbn\|Drorbn]] 16:25, 15 November 2010 (EST)

Drorbn at 21:25, 15 November 2010

2010-11-15T21:25:23Z

← Older revision		Revision as of 17:25, 15 November 2010
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	* Dear Prof. Bar-Natan, what is Problem 1?		* Dear Prof. Bar-Natan, what is Problem 1?
			** Dear Dror, just wait and see. At the moment I have no clue. [[User:Drorbn\|Drorbn]] 16:25, 15 November 2010 (EST)

Drorbn at 21:23, 15 November 2010

2010-11-15T21:23:59Z

← Older revision		Revision as of 17:23, 15 November 2010
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	'''Problem 1.'''		'''Problem 1.'''

			* Dear Prof. Bar-Natan, what is Problem 1?

Drorbn at 21:22, 15 November 2010

2010-11-15T21:22:09Z

New page

{{10-1100/Navigation}}
{{In Preparation}}

This assignment is due at class time on Tuesday, November 30, 2010.

===Solve the following questions===

'''Problem 1.'''