Maximal ideals of a quotien ring

[R.P.F.]

Homework Statement

I am try to prove :
Let R be a ring, I be a ideal of R.
Then N is a maximal in R/I if and only if N=M/I where M is a maximal ideal in R that contains I.

Homework Equations

The Attempt at a Solution

First I'm not 100% sure that the statement is true, but I'm trying to prove it.
So N is maximal in R/I, then R/I/N is a field which means I/N is maximal in R. This is as far as I get.
Thanks!

 

[micromass]

This statement is 100% correct. So don't worry about it.

Have you heard of the fourth isomorphism theorem (you probably didn't call it that). It states that there is a bijective correspondance between ideals of R that contain I and ideals of R/I.
In particular, if J is an ideal of R/I, then J=J'/I for some ideal J' of R.

This is the thing you have to use to prove this question. If you didn't see it, then perhaps you could try to prove it...

 

[R.P.F.]

This statement is 100% correct. So don't worry about it.

Have you heard of the fourth isomorphism theorem (you probably didn't call it that). It states that there is a bijective correspondance between ideals of R that contain I and ideals of R/I.
In particular, if J is an ideal of R/I, then J=J'/I for some ideal J' of R.

This is the thing you have to use to prove this question. If you didn't see it, then perhaps you could try to prove it...

Yay I figured it out using your hint! Thanks a lot!

 

[micromass]

Well, x² isn't an irreducible polynomial of R[x], since x²=x.x
Thus R[x]/(x²) is not a field.