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

In Preparation

The information below is preliminary and cannot be trusted! (v)

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 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 \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 $n$, 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^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 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 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 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 1+13i} , and express it as a linear combination of these two elements.

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

Problem 7. (Dummit and Foote) Prove that the quotient of a PID by a prime ideal is again a PID.