Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the following natural homeomorphisms $Spec((1+I)^{-1}R)\cong\{p\in Spec(R): 1+i\not\in p\}$ and $Spec(R/J)\cong\{p\in Spec(R): J\subseteq p\}$.
I am looking for (necessary or(and) sufficient)conditions on $I$ and $J$ under which $$\{p\in Spec(R): 1+i\not\in p\}\subseteq \{p\in Spec(R): J\subseteq p\}.$$ Or under the above homeomorphisms $$Spec((1+I)^{-1}R) \subseteq Spec(R/J).$$
Note that $Spec(S)$ is the set of all prime ideals of a ring $S$.
