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Prove the Lattice Isomorphism Theorem for Rings. That is, if I is an ideal of a...

Prove the Lattice Isomorphism Theorem for Rings. That is, if I is an ideal of a ring R, show that the correspondence A ↔ A/I is an inclusion preserving bijections between the set of subrings A ⊂ R that contain I and the set of subrings of R/I.

Furthermore, show that A (a subring containing I) is an ideal of R if and only if A/I is an ideal of R/I.

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