 Solution

Suppose $R$ is an integral domain and $(a)=(b)$. If $a=0$, then $(b)=0$, so $b=0$ and $a=1\cdot b$.

Otherwise membership in the equal principal ideals supplies $r,s\in R$ with $a=rb$ and $b=sa$. Substituting gives $a=rsa$, or $(1-rs)a=0$. There are no nonzero zero divisors in an integral domain, and $a\ne 0$, so $rs=1$. Hence $r$ is a unit with inverse $s$, and $a=rb$ makes $a$ and $b$ associate elements. We have proved