Explanation behind a 'non-irreducible' as a product of irreducible

Question

I am currently looking at a proof behind the theorem:

Every Principal ideal domain is a unique factorisation domain.

In a part of the proof:

"Let $a_{0}$ be an element in the principal ideal domain written as $a_{0}=p_{1}c_{1}$ where $p_{1}$ is irreducible and $c_{1}$ is not a unit. If $c_{1}$ is not irreducible then we can write $c_{1}=p_{2}c_{2}$ where $p_{2}$ is irreducible and $c_{2}$ is not a unit..."

Can someone shed some light as to how an element that is not an irreducible be written as a product of an element that is an irreducible? I've kept going back to the definition for irreducible but am none the wiser.

Thanks in advance.

Answer 1

The author is using the fact that each nonzero nonunit in a PID has at least one irreducible factor (which is the claim in the first paragraph of the attached screenshot).

Answer 2

• If $R$ is principal ideal domain, then by definition every ideal is generated by one element.

• Every maximal ideal is prime.

• in integral domain every prime element is irreducible

Now for arbitrary $a\in R$ that is not unit, there exists maximal ideal such, that $(a)\subset M$, and $\exists m\in R:\ (m) = M$.

$m$ is irreductible, and $a\in (m)$, therefore $\exists t\in R:\ mt=a$.