# A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

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 $$S$$.