= Correspondence theorem for ideals
= Ideal correspondence theorem
{synonym}
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.
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