Basic Ring Theory - 8

[Tsakanikas Nickos]

Let \( A \) be an integral domain and let \( \mathbb{K} \) be its field of fractions. Show that the following are equivalent:

1. \( A \) is a valuation ring of \( \mathbb{K} \).
2. If \( \mathfrak{a},\mathfrak{b} \) are two ideals of \( A \), then either \( \mathfrak{a} \subseteq \mathfrak{b} \) or \( \mathfrak{a}\supseteq \mathfrak{b} \).