Let be $$R= \mathbb{Z} [ \sqrt{-6} ]= \{a+b \sqrt{-6} | a,b \in \mathbb{Z} \} $$
I want to prove following properties from a book:
a) Any Divisor of $2$ is either a unit or associated to $2$. And also any divisor of $\sqrt{-6} $ is either a unit or associated to $\sqrt{-6} $
b) Find Elements $x,y \in R $ so that $ \sqrt{-6} \nmid x $ and $ \sqrt{-6} \nmid y $ but $ \sqrt{-6} \mid xy $
c) $1$ is the $\gcd$ of $2$ and $ \sqrt{-6} $ but there don't exist $a,b \in R $ so that $ 2a + \sqrt{-6} b =1 $
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$ a|b \leftrightarrow an=b $ , $ n \in \mathbb{Z} $
$a$ and $b$ associated $ \leftrightarrow $ if there exists a $\epsilon$, so that $ b=a\epsilon $ $\leftrightarrow $ $ b \mid a $ & $ a \mid b $
I dont know what elements I can take from $ R$.. I appreciate any help!
For (b), $n(1\pm\sqrt{-6})$ is not divisible by $\sqrt{-6}$ except where $n$ is a multiple of $6$, but consider the product $(n(1+\sqrt{-6}))\cdot(m(1-\sqrt{-6}))$ when $n$ is a multiple of $2$ and $m$ is a multiple of $3$.
We get this type of example whenever (the magnitude of) the radicand is composite.