**189-571B:** Higher Algebra II

## Assignment 2. Due: Wednesday, February 7.

**1.** Show that a UFD is integrally closed. Is the converse true?
Explain.

**2. ** Which of the following rings are Dedekind domains?
Justify your answers.

(a) $R= {\bf Z}[x]/(x^2+9)$;

(b) $R= {\bf Z}[x]/(9x^2+1)$;

(c) $R= F[x,y]$ where $F$ is a field.

(d) $R = F[x,y]/(y^2-x^3-1)$, where $F$ is a field.

**3.**
Problem 48, page 450 of *Basic Algebra*.

**4.**
Problem 49, page 450 of * Basic Algebra*.

**5.**
Problem 50, page 450 of * Basic Algebra*.

For the following questions, let
$R$ be a Noetherian local ring which is an integral domain,
with maximal ideal equal to ${\frak m}$.
Let $F$ denote the fraction field of $R$.
The goal of these questions is to show that if $R$ is a Dedekind domain, then
it is a PID (and hence, a discrete valuation ring).

**6.**
Show that, if $I$ is any ideal of $R$, then $I$ contains
${\frak m}^n$ for a large enough integer $n$.

**7.** Show that there is a principal ideal $(t)$ in ${\frak m}$
which is not properly contained in any other
principal ideal in ${\frak m}$.

**8. ** Use the results of questions 6 and 7 to show that there is an
element $b\in R$ for which:

(a) the element $b/t$ belongs to $F-R$, and

(b) The set
$b/t \cdot {\frak m}$ is contained in $R$.

** 9. ** With notations and assumptions as in the questions 6-8, show that
$b/t \cdot {\frak m} = R$ if $R$ is integrally closed.
(Hint: otherwise, use the fact that
$b/t \cdot {\frak m} \subset {\frak m}$, and the assumtion that $R$ is integrally closed, to derive a contradiction.)

**10.**
Use the fact that $b/t \cdot {\frak m} = R$ to conclude
that $(t)$ contains ${\frak m}$.
Conclude that ${\frak m}$ is a principal ideal and hence that $R$ is a
PID.