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Irreducible elements

  1. Mar 7, 2005 #1
    Here’s an interesting question…
    Let R be a commutative ring and ‘a’ an element in R. If the principal ideal Ra is a maximal ideal of R then show that ‘a’ is an irreducible element.
    If a is prime, this is pretty obvious…if a is not prime, then we say a= bc for some b,c in R. Now we need to show that either of them is a unit. I can’t imagine how… :frown:
  2. jcsd
  3. Mar 8, 2005 #2


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    If a is reducible, try finding a bigger ideal than (a). Maximal ideals are always prime ideals!
  4. Mar 8, 2005 #3


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    the usual definition of a prime element in a commutative ring with identity, is that it generates a prime ideal.
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