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Week of...

Notes and Links

1

Sep 13

About This Class, Tuesday  Non Commutative Gaussian Elimination, Thursday

2

Sep 20

Tuesday  Homomorphisms and Normal Groups, Thursday  Isomorphism Theorems

3

Sep 27

Class Photo, HW1, HW1 solution

4

Oct 4


5

Oct 11

HW2, HW2 solution

6

Oct 18


7

Oct 25

Term Test

8

Nov 1

HW3, HW3 solution

9

Nov 8

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

10

Nov 15

HW4, HW4 solution

11

Nov 22


12

Nov 29

HW5, HW5 solution

13

Dec 6

Boxing Day Handout, see also December 2010 Schedule. Some Class Notes

F

Dec 13

Final exam, Tuesday December 14 101, Bahen 6183

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See Non Commutative Gaussian Elimination


This assignment is due at class time on Tuesday, November 30, 2010.
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. (Withdrawn, do not submit) Show that the quotient ring ${\mathbb {Q} }[x,y]/\langle x^{2}+y^{2}1\rangle$ is not a UFD.