Let $\mathbb{Z}[T_1,...,T_n]$ be the ring of polynomials over $\mathbb{Z}$, and $\mathbb{F}_p[T_1,...,T_n]$ be the ring of polynomials over $\mathbb{F}_p$, with the canonical projection
$p:\mathbb{Z}[T_1,...,T_n]\rightarrow \mathbb{F}_p[T_1,...,T_n].$
Let $I=(f_1,...,f_r)$ be an ideal of $\mathbb{Z}[T_1,...,T_n]$, then we define $\overline{I}:=(\overline{f_1},...,\overline{f_r})=I^e=p(I)$ be the reduction ideal of $I$ in $\mathbb{F}_p[T_1,...,T_n]$.
My question is, for ideals $I$ and $J$ of $\mathbb{Z}[T_1,...,T_n]$, in what kind of condition, we have $\overline{I\cap J}=\overline{I}\cap \overline{J}$?
(I think the above equality can not be true in general, however if we ask some condition of the generator of $I$ and $J$, for example with trivial content of grobner basis, maybe some good thing happen?)