Clarification on Exponents in Prime Factorization of Ideals in Dedekind Domains and Number Fields

Question

Let $R$ be a Dedekind domain and $I$ a proper ideal. Then I know $I$ can be expressed uniquely as a finite product of prime ideals: $$ I = \prod_{\mathfrak{p} \text{ prime}} \mathfrak{p}^{n_{\mathfrak{p}}}, $$ for some non-negative integers $n_{\mathfrak{p}}$. When talking about number fields, they are in particular Dedekind domains. Let $K$ be a number field. Then every prime ideal $\mathfrak{p}$ on $K$ defines a valuation $v_{\mathfrak{p}}$ such that $v_{\mathfrak{p}}(\alpha) = n_{\mathfrak{p}}$ where this integer is the highest exponent of the prime $\mathfrak{p}$ in the prime factorzation of the ideal $(\alpha)$. This is, off course, as I understand it, but then I saw this post: Extending the p-adic valuation and in there they say that $n_{\mathfrak{p}}$ can be negative. Why? I mean, I can express any $I$ as a finite product of distinct prime ideals, but why can some of them have negative powers? Or am I confused maybe? But in that case, what about the ring of algebraic integers of $K$? Do they all have non-negative valuation? Thanks in advance; maybe it is a potential silly confusion.