Prove that any element in a ring A can be written as a product of irreducible but A is not a UFD

Question

Let A = $Z$[$\sqrt{10}$] = {a+b$\sqrt{10}$$\mid$a,b $\in$ $Z$}.

Prove that any element in A an be written as a product of irreducible, but A is not a UFD.

Also a small question, I've proved that $4$+$\sqrt{10}$ and $4$-$\sqrt{10}$ are irreducible, but how to show that they cannot be associates?

Thank you so much!

Answer 1

Since $\mathbb{Z}[\sqrt{10}]$ is Noetherian, every element can be written as a finite product of irreducible elements. For a proof, see the arguments here:

Every element of $Z (\sqrt{-5})$ is factorable into irreducible factors.

The other part, the main part, is proved here:

Proving $\mathbb{Z}[\sqrt {10}]$ is not a UFD