Let $K$ be a finite extension of the field of rational numbers. Let $A$ be the ring of algebraic integers in $K$. Let $I$ be a non-zero ideal of $A$. Let $\alpha$ be a non-zero element of $K$ which is relatively prime to $I$.
Are there algebraic integers $\beta$, $\gamma$ in $A$ with the following properties?
(1) $\alpha = \beta/\gamma$.
(2) $\beta$ and $\gamma$ are relatively prime to $I$.
EDIT(Mar.2,2013) Here is a generalization of this question.