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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 10:3012:30 (Dror), 5PM7PM @ 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

MondayTuesday is November Break, One Theorem, Two Corollaries, Four Weeks

10

Nov 14

HW4, HW 4 Solutions

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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

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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 nonzero $a,b\in R$.
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 $R$.
 Prove that if $R$ is commutative then $\eta (R)$ is an ideal.
 Give an example of a noncommutative 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.