M maximal iff M ¯ is maximalWe have a ring R and I an ideal of R. Let M be an ideal of R containing I. Let M ¯ be M/I and R ¯ be R/I. Prove that M is maximal if and only if M ¯ is maximal.I think I get the general idea, that M not maximal means there exists a bigger M′ containing M and so M ′ / Icontains M / I. How do I formalize this idea? And how do I show the other direction?

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M  maximal iff       M    ¯   is maximalWe have a ring R and I an ideal of R. Let   M be an ideal of R containing I. Let       M    ¯   be M/I and       R    ¯   be R/I. Prove that   M is maximal if and only if       M    ¯   is maximal.I think I get the general idea, that   M not maximal means there exists a bigger   M′ containing   M and so       M    ′        /    Icontains   M      /    I. How do I formalize this idea? And how do I show the other direction?

Mylee Mcintyre · Accepted Answer

By the third isomorphy theorem you haveR/M≅(R/I)/(M/I).Each of this quotients is a field if and only if M is maximal, and if and only if M/I is maximal.

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