# Homework Help: Nonzero nonunit non-product-of-irreducibles is reducible?

1. May 6, 2012

### julypraise

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. May 6, 2012

### jgens

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

3. May 7, 2012

### julypraise

Thanks a lot!