11-1100/Homework Assignment 4 - Revision history

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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.

		⚫	'''Problem 6.''' (Hard!) Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.
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⚫	'''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 (~~Withdrawn, do not submit~~) Show that the quotient ring 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 00:08, 15 November 2011

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This assignment is due at class time on Tuesday, November 29, 2011.

===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 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.

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'''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 (Withdrawn, do not submit) Show that the quotient ring 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.