= Associator ring
{title2=$(a)=(b)\Longrightarrow a=ub\text{ for some }u\in R^\times$}
An associator ring is a <ring> in which equality of the <principal ideals> generated by two elements forces them to be <associate elements>: $(a)=(b)$ implies $a=ub$ for a <unit in a ring> $u$. The reverse implication always holds, since $a=ub$ and $b=u^{-1}a$ give both <ideal> inclusions. The property separates equality of generated <ideals> from equality of generators up to an invertible factor.
Every <integral domain> is an associator ring. If $a\ne0$ and $(a)=(b)$, write $a=rb$ and $b=sa$. Then $(1-rs)a=0$, so cancellation gives $rs=1$ and $a=rb$ with $r$ a <unit>. The zero case is immediate. The converse fails: $\mathbb Z/4\mathbb Z$ has generator classes $\{0\}$, $\{2\}$ and $\{1,3\}$ for its three <ideals>, and $1,3$ differ by a <unit>, although $2$ is a nonzero <zero divisor>.
Finite <direct product of rings> preserve this property. Equality of <principal ideals> is coordinatewise, and a tuple of <units> is a <unit>. Every <quotient ring> of a <principal ideal domain> is also an associator ring: first, modulo a prime power $p^k$, every nonzero class is a power $p^e$, $e<k$, times a <unit>, and distinct powers generate distinct <ideals>. Next factor a nonzero nonunit modulus into prime powers and apply the <Chinese remainder theorem>. The zero modulus gives the original <integral domain>, while a unit modulus gives the <zero ring>.
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