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Maximal Ideal/Ring homomorphism question

  1. May 9, 2012 #1
    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.
     
  2. jcsd
  3. May 9, 2012 #2

    jgens

    User Avatar
    Gold Member

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