Confusion about the definition of ideal (of ring of algebraic integers)

Question

I am studying algebraic number theory and am confused about the following lemma. We prove that if $I \subset O_K$ a non-zero ideal then $$\textrm{disc}(I) = \textrm{disc}(O_K)\cdot N(I)$$ Then somewhere later on the author considers the case where $K = \mathbb{Q}(\alpha)$ and wants to find the index of $\mathbb{Z}[\alpha]$ in $O_K$. So he substitutes $\mathbb{Z}[\alpha]$ for $I$ in the above formula and calculates it straightforwardly. However I am confused as $\mathbb{Z}[\alpha]$ does not seem to be an ideal of $O_K$. I.e. it does not absorb products. I am missing something here?