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Nonzero nonunit non-product-of-irreducibles is reducible?

  1. May 6, 2012 #1
    1. The problem statement, all variables and given/known data
    Claim
    Let R be an integral domain. Then a nonzero nonunit element in R that is not a product of irreducible elements is reducible.

    Is this claim true?


    2. Relevant equations



    3. The attempt at a solution
    By definition, a product of irreducibles is reducible because irreducibles are not units. This implies if nonzero nonunit x is irreducible then it is not a product of irreducible elements.

    I've tried to prove the above claim by contradiction supposing nonzero nonunit x is both not a product of irreducibles and x is irreducible. But by the above observation, I couldn't; the fact that x is irreducible and the fact that x is nonzero nonunit and not a product of irreducibles are consistnet.

    Logically, the claim asserts that if x is nonzero nonunit and not a product of irreducibles then x is reducible. Let's call this statement P -> Q. And the above observation yields, kinda, that ~P -> Q. But this this yields every nonzero nonunit element in R is reducible, which is absurd.

    So the claim should be false?
     
  2. jcsd
  3. May 6, 2012 #2

    jgens

    User Avatar
    Gold Member

    The claim is false. The integers provide a good counter-example.
     
  4. May 7, 2012 #3
    Thanks a lot!
     
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