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 $R$ is a PID iff it is a UFD in which $\gcd(a,b)\in \langle a,b\rangle$ for every non-zero $a,b\in R$.

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

1. Prove that if $R$ is commutative then $\eta (R)$ is an ideal.
2. Give an example of a non-commutative ring $R$ in which $\eta (R)$ is not an ideal.

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

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

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

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