Let $H$ be a Hopf algebra, and $(M,\triangleleft)$ a $H$-module. Now for $m \in M$, and $h \in H$, then is it true in general that
$$
m \triangleleft h = 0 ~~~ \implies ~~~~m \triangleleft S(h) = 0? 
$$
If not, do there exist conditions on $H$ or $M$ under which it is true. 


Thanks in advance guys!



Zero Actions on a Hopf Module Preserved Under the AntiPode?
Not true. All you need is a right ideal, not closed under $S$. Then you will a cyclic module with a counterexample. The group algebra of $C\_3$ over a field with a primitive cubic root of 1 will be an example...


I cannot think of any conditions beyond obvious ones...


