Basic Ring Theory - 15

[Tsakanikas Nickos]

Let \( A \) be a noetherian, local, integral domain of dimension \( 1 \) with maximal ideal \( \mathfrak{m} \). Show that the following are equivalent:

1. \( A \) is a discrete valuation ring.
2. \( A \) is integrally closed (i.e. a normal ring).
3. \( A \) is a regular local ring.
4. \( \mathfrak{m} \) is a principal ideal.