Irreducible elements and Associates

Question

Show that, in a domain, every associate of an atom is an atom.

An atom is the same thing as an irreducible element.

I think these two facts will be important to prove this statement:

1. A nonunit is an atom if and only if it cannot be written as a product of two nonzero nonunits

2. Two elements are associates if and only if one is a unit multiple of the other.

I just need help with the actual proof writing. I don't know how to convert this information into a nice flowing proof. Any advice would be appreciated!

I'm also a little confused because of this link Irreducible elements are not associates

Answer 1

By $(2)$ an associate of an atom $\,p\,$ must be a unit multiple $\,up.\\$ Ifthis associate were reducible then $\,up = ab\,$ so $\,p = (u^{-1}a) b\,$ is reducible, contradiction.

Generally since the relation of divisibility is preserved by unit scalings, so too are pure divisibility properties such are irreducibility, primality. etc.

Answer 2

Let $p$ be an atom and $u$ an unit. If $pu$ weren't an atom, we'd have $pu=bc$, and $p=b(cu^{-1})$, a contradiction.