$R$ integral domain : $u\in R^*, a \text{ is prime} \iff au \text{ is prime}$

Question

$R$ integral domain : $u\in R^*,\; a \text{ is prime} \iff au \text{ is prime}$

I started by looking at $auu^{-1}$. What should I do next? I'd be glad for help.

Note: $u \in R^*$ meaning is $u$ is invertible.

Answer 1

Suppose $au\mid bc$. Then, there exists $k$ such that $auk=bc$. Therefore, $a\mid bc$. Since $a$ is prime, $a$ divides $b$ or $c$. Then, $au$ divides $bu$ or $cu$.

If $au$ divides $bu$ then $auj=bu$ for some $j$, and $auju^-1=b$; thus, $au\mid b$.

A similar reasoning gives that if $au$ divides $cu$ then $au$ divides $c$.

We supposed that $au\mid bc$ and we have concluded that $au$ divides $b$ or $c$. Therefore, $au$ is prime.

Answer 2

$$a\;\;\text{is prime}\;\;\iff\;\; aR=\langle a\rangle\;\;\text{is a prime ideal}$$

and the claim follows at once observing that $\;\langle a\rangle = \langle au \rangle\;$