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.