Categories

Exponents of Prime Ideals in Number Fields

I have a question about “factoring out” common prime ideal factors in a number field.

Let $$K$$ be a number field with ring of integers $$\mathcal{O}_K$$. Let $$\mathfrak{p}$$ be a prime ideal of $$\mathcal{O}_K$$.

Now let $$k \in K$$. I know I can write $$k = a/b$$ for $$a,b \in \mathcal{O}_K$$. Now consider the fractional ideal $$k\mathcal{O}_K$$. Let $$v_\mathfrak{p}(x)$$ be the exponent that $$\mathfrak{p}$$ appears to in the prime factorization of this fractional ideal. We of course have that $$v_\mathfrak{p}(x) = v_\mathfrak{p}(a)- v_\mathfrak{p}(b)$$. Now if $$v_\mathfrak{p}(x) \geq 0$$, is it always possible to choose $$a,b$$ in such a way so that $$v_\mathfrak{p}(b) = 0$$. So in a way, are we able to “factor out” common factors in their prime decomposition?