Maximal Ideal/Ring homomorphism question

[Zoe-b]

Homework Statement

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

Homework 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.

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.

 

[jgens]

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.

An embedding is necessarily injective. Your mapping is not injective so it is not an embedding. For this problem consider the inclusion homomorphism $\mathbb{Z} \rightarrow \mathbb{R}$ and recall that the only maximal ideal in a field is $\{0\}$.