11-1100/Homework Assignment 4

DBN: Classes: 2011-12: 11-1100/Navigation # Week of... Notes and Links Additions to the MAT 1100 web site no longer count towards good deed points 1 Sep 12 About This Class, Tuesday - Non Commutative Gaussian Elimination, Thursday - NCGE completed, the category of groups, images and kernels. 2 Sep 19 I'll be in Strasbourg, class was be taught by Paul Selick. See my summary. 3 Sep 26 Class Photo, HW1, HW1 Submissions, HW 1 Solutions 4 Oct 3 The Simplicity of the Alternating Groups 5 Oct 10 HW2, HW 2 Solutions 6 Oct 17 Groups of Order 60 and 84 7 Oct 24 Extra office hours: Monday 10:30-12:30 (Dror), 5PM-7PM @ Huron 1028 (Stephen). Term Test on Tuesday. Summary for the midterm exam. Term Test - Sample Solutions 8 Oct 31 HW3, HW 3 Solutions 9 Nov 7 Monday-Tuesday is November Break, One Theorem, Two Corollaries, Four Weeks 10 Nov 14 HW4, HW 4 Solutions 11 Nov 21 12 Nov 28 HW5 and last week's schedule 13 Dec 5 Tuesday - the Jordan form; UofT Fall Semester ends Wednesday; our Final Exam took place on Friday Register of Good Deeds Add your name / see who's in! See Non Commutative Gaussian Elimination

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

Solve the following questions

Problem 1. Prove that a ring [math]\displaystyle{ R }[/math] is a PID iff it is a UFD in which [math]\displaystyle{ \gcd(a,b)\in\langle a, b\rangle }[/math] for every non-zero [math]\displaystyle{ a,b\in R }[/math].

Problem 2. (Selick) In a ring [math]\displaystyle{ R }[/math], and element [math]\displaystyle{ x }[/math] is called "nilpotent" if for some positive [math]\displaystyle{ n }[/math], [math]\displaystyle{ x^n=0 }[/math]. Let [math]\displaystyle{ \eta(R) }[/math] be the set of all nilpotent elements of [math]\displaystyle{ R }[/math].

1. Prove that if [math]\displaystyle{ R }[/math] is commutative then [math]\displaystyle{ \eta(R) }[/math] is an ideal.
2. Give an example of a non-commutative ring [math]\displaystyle{ R }[/math] in which [math]\displaystyle{ \eta(R) }[/math] is not an ideal.

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

Problem 4. (Lang) Show that the ring [math]\displaystyle{ {\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]\displaystyle{ {\mathbb Z}[i] }[/math], find the greatest common divisor of [math]\displaystyle{ 85 }[/math] and [math]\displaystyle{ 1+13i }[/math], and express it as a linear combination of these two elements.

Problem 6. (Hard!) Show that the quotient ring [math]\displaystyle{ {\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle }[/math] is not a UFD.