Intersections of ideals in group rings obtained by extension-of-scalars

Question

I stumbled upon the following unproved claim in a paper I'm reading and can't quite see why it's true. Suppose I have an ideal $I$ of the group ring $\mathbb{Z}[G]$ where $G$ is a finite abelian group. Let $\mathbb{Z}_p[G]I$ be the ideal of the group ring $\mathbb{Z}_p[G]$ generated by $I$. It is clear that $I\subset \mathbb{Z}[G]\cap \mathbb{Z}_p[G]I$ and that this intersection is often proper. For instance let $G$ be cyclic of order two with generator $\sigma$ and let $I=(2+\sigma)$. Then for $p\neq 5$, $1=(2+\sigma)\frac{1}{5}(2-\sigma) \in \mathbb{Z}_p[G]I$. However, $(2+\sigma) \mathbb{Z}[G]\cap \mathbb{Z}_5[G]I$. Is it true that $I=\bigcap_p(\mathbb{Z}[G] \cap \mathbb{Z}_p[G]I)$? The paper I'm reading implies this is trivial but I'm having a difficult time verifying it. Any hints would be much appreciated.