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julypraise
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Homework Statement
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?
Homework Equations
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?