Faithful Cyclic Modules for Enveloping Algebras and Sklyanin Algebras
  Let $U$ be the enveloping algebra of a finite dimensional nonabelian Lie algebra $\mathfrak{g}$ over a field of characteristic zero. We show that there is an open nonempty open subset $X$ of $U_1 = \mathfrak{g}\oplus K$ such that   $U/Ux$ is faithful for all $x \in X$. We prove similar results for homogenized enveloping algebras and for the three dimensional   Sklyanin algebras at points of infinite order. It would be interesting to know if there is a common generalization of these results. 
Ian M. Musson