= 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 $a\ne0$, so $rs=1$. Hence $r$ is a <unit> with inverse $s$, and $a=rb$ makes $a$ and $b$ <associate elements>. We have proved
$$
\boxed{\text{Every integral domain is an associator ring.}}
$$
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