(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

So I have a question that says:

Let T:R -> S be a ring homomorphism, show that if J is a prime ideal of S, then

T^{-1}(J) := { r in R s.t. T(r) is in J)

is a prime ideal of R. (I've done this bit)

It then says:

Give an example where J is maximal but T^{-1}(J) is not maximal, hint: consider a suitable embedding T of a ring into a field

2. Relevant equations

First thing that doesn't really help is that I'm not so clear of what an 'embedding of a ring into a field' actually means in the first place. This is a phrase that crops up but has never been properly defined in my course.

3. The attempt at a solution

Ok so if I let R = integers and S = integers mod 7, then I think T taking a in Z to its equivalence class mod 7 defines an embedding of the integers into the field Z mod 7. However the only ideals in Z mod 7 are the whole field, and {0}. The pre-image of the whole field is clearly the whole of Z, whereas the pre-image of {0} is the set 7Z which is also a maximal ideal since 7 is prime. Bit confused, have tried a few other examples but can't get anything to work/understand the hint.

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# Homework Help: Maximal Ideal/Ring homomorphism question

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