Let $R$ be a finite commutative local ring with identity. Assume that every ideal in $R$ is principal.
Let $u$ and $v$ be units in $R$ and let $z\neq 0$ be a zero divisor. I think that $uz=vz$ implies $u=v$, but I have troubles in finding a proof.
In fact if the maximal ideal of $R$ is generated by $\alpha$ then we can assume that $z=w\alpha ^t $, for a unit $w$.
The claim is wrong: $R=\mathbb Z/9\mathbb Z$, $u=4$, $v=1$, and $z=3$.