# 11-1100/Homework Assignment 4

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

### 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 $a,b\in R$.

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 $R$.

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