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?