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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!

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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...

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