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

[julypraise]

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?

 

[jgens]

The claim is false. The integers provide a good counter-example.

 

[julypraise]

Thanks a lot!