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Prime ideals in ring theory

  1. Apr 27, 2010 #1
    1. The problem statement, all variables and given/known data

    Given a commutative ring with unity, show that if every ideal is prime than the ring is a field.

    2. Relevant equations



    3. The attempt at a solution

    I think that I can show that a ring is a field iff it has no nontrivial ideals. So I guess I need to show that if a ring has only prime ideals than these ideals must be trivial. I'm not sure how to do this though.
     
  2. jcsd
  3. Apr 27, 2010 #2
    Can you show that the ring must be an integral domain? (Hint: Consider the ideal that consists just of the zero element.) Next show that every non-zero element is a unit to complete the proof. (Hint: Let r be a non-zero element. Consider the ideal generated by [itex]r^2[/itex].)

    Petek
     
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