If $R$ is Noetherian and $I$ an ideal why ideals in $R/I$ correspond to ideals in $R$ containing $I$?
I read that if $J\supseteq I$ is an ideal of $R$ then each coset of $I$ is either contained in $J$ or disjoint from $J$ , and thus $J$ maps directly to a subset of R/I via the canonical projection homomorphism but i dont understand at all the part that says then each coset of $I$ is either contained in $J$ or disjoint from $J$ can you help me to understand that? thank you
