= Solution
If $a=ub$ with $u$ a <unit in a ring>, then $a$ lies in the <principal ideal> generated by $b$. This gives $(a)\subseteq(b)$. Since $u$ has a <multiplicative inverse>, $b=u^{-1}a$, giving the reverse inclusion. Therefore
$$
\boxed{a=ub,\ u\in R^\times\quad\Longrightarrow\quad(a)=(b).}
$$
Thus <associate elements> always generate the same <principal ideal>; the <associator ring> condition asks whether equality of these <ideals> also forces the generators to differ by a <unit>.
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