10-1100/Homework Assignment 4

DBN: Classes: 2010-11: 10-1100/Navigation Additions to the MAT 1100 web site no longer count towards good deed points # 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 Monday-Tuesday 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 10-1, Bahen 6183 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 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 non-zero Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b\in R} .

Problem 2. (Selick) In a ring $R$, and element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. Show that the quotient ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle} is not a UFD.