# 11-1100/Homework Assignment 4

This assignment is due at class time on Tuesday, November 29, 2011.

### Solve the following questions

Problem 1. Prove that a ring ${\displaystyle R}$ is a PID iff it is a UFD in which ${\displaystyle \gcd(a,b)\in \langle a,b\rangle }$ for every non-zero ${\displaystyle a,b\in R}$.

Problem 2. (Selick) In a ring ${\displaystyle R}$, and element ${\displaystyle x}$ is called "nilpotent" if for some positive ${\displaystyle n}$, ${\displaystyle x^{n}=0}$. Let ${\displaystyle \eta (R)}$ be the set of all nilpotent elements of ${\displaystyle R}$.

1. Prove that if ${\displaystyle R}$ is commutative then ${\displaystyle \eta (R)}$ is an ideal.
2. Give an example of a non-commutative ring ${\displaystyle R}$ in which ${\displaystyle \eta (R)}$ is not an ideal.

Problem 3. (comprehensive exam, 2009) Let ${\displaystyle A}$ be a commutative ring. Show that a polynomial ${\displaystyle f\in A[x]}$ is invertible in ${\displaystyle A[x]}$ iff its constant term is invertible in ${\displaystyle A}$ and the rest of its coefficients are nilpotent.

Problem 4. (Lang) Show that the ring ${\displaystyle {\mathbb {Z} }[i]=\{a+ib\colon a,b\in {\mathbb {Z} }\}\subset {\mathbb {C} }}$ is a PID and hence a UFD. What are the units of that ring?

Problem 5. (Dummit and Foote) In ${\displaystyle {\mathbb {Z} }[i]}$, find the greatest common divisor of ${\displaystyle 85}$ and ${\displaystyle 1+13i}$, and express it as a linear combination of these two elements.

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