||Notes and Links
|Additions to the MAT 1100 web site no longer count towards good deed points
||About This Class, Tuesday - Non Commutative Gaussian Elimination, Thursday - NCGE completed, the category of groups, images and kernels.
||I'll be in Strasbourg, class was be taught by Paul Selick. See my summary.
||Class Photo, HW1, HW1 Submissions, HW 1 Solutions
|| The Simplicity of the Alternating Groups
||HW2, HW 2 Solutions
||Groups of Order 60 and 84
||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
||HW3, HW 3 Solutions
||Monday-Tuesday is November Break, One Theorem, Two Corollaries, Four Weeks
||HW4, HW 4 Solutions
||HW5 and last week's schedule
||Tuesday - the Jordan form; UofT Fall Semester ends Wednesday; our Final Exam took place on Friday
|Register of Good Deeds
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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 is a PID iff it is a UFD in which for every non-zero .
Problem 2. (Selick) In a ring , and element is called "nilpotent" if for some positive , . Let be the set of all nilpotent elements of .
- Prove that if is commutative then is an ideal.
- Give an example of a non-commutative ring in which is not an ideal.
Problem 3. (comprehensive exam, 2009) Let be a commutative ring. Show that a polynomial is invertible in iff its constant term is invertible in and the rest of its coefficients are nilpotent.
Problem 4. (Lang) Show that the ring is a PID and hence a UFD. What are the units of that ring?
Problem 5. (Dummit and Foote) In , find the greatest common divisor of and , and express it as a linear combination of these two elements.
Problem 6. (Hard!) Show that the quotient ring is not a UFD.