[Math] Irreducible elements and Associates

integral-domainproof-writingring-theory

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

Best Answer

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.

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