Let $R$ be an integral domain and $x$ is non zero element in $R$. Show that $R\cong Rx$ (as rings) if and only if $x$ is an unit.
I know that $x\in R$ is a unit if there exist $y\in R$ such that $xy=1$. What the hint to proof this theorem?
I try to prove like this. Given $R\cong Rx$, so there exist an isomorphism $\theta : R\to Rx$. Now I confuse how to define the $\theta$ mapping. I can't associate with the unit element definition.
Let $\phi$ be the ring isomorphism $R\to Rx$. It's obvious that $\phi(1)$ is a nonzero idempotent of $Rx$, that is, $\phi(1)^2=\phi(1)$.
Suppose $\phi(1)=rx$. Since the image is also an integral domain, you can cancel the expression $rxrx=rx$ to get $rx=1$. At that point, you would have discovered $x$ is a unit.