 Correspondence theorem for ideals

For a commutative ring $R$ and an ideal $I$, the maps $J\mapsto J/I$ and $H\mapsto q^{-1}(H)$ give inverse inclusion-preserving bijections between ideals $J\supseteq I$ of $R$ and ideals $H$ of the quotient ring $R/I$, where $q:R\to R/I$ is the quotient map. The preimage is an ideal containing the kernel $I$. Conversely, $q^{-1}(q(J))=J$ for $J\supseteq I$, and surjectivity of $q$ gives $q(q^{-1}(H))=H$. These identities prove the claim.