S.Dascalescu, C.Nastasescu and S.Raianu define the action of a Hopf algebra $H$ on an (associative) algebra $A$ as a map $H\times A\owns (h,a)\mapsto h\cdot a\in A$ which
- is an action of $H$ on $A$ as an algebra on a vector space and
- satisfies two supplementary conditions: $$ h\cdot(a\cdot b)=\sum(h_1\cdot a)\cdot (h_2\cdot b),\qquad h\cdot 1_A=\varepsilon(h)\cdot 1_A. $$
I think that the kernel of such an action, i.e. the set
$$
I=\{h\in H:\quad \forall a\in A\quad h\cdot a=0\},
$$
must be a biideal in $H$ as in a bialgebra, i.e.
$$
\Delta(I)\subseteq I\otimes H+H\otimes I,
$$
but this is quite far from me, I have no idea how people prove this. Can anybody enlighten me? Thank you.
P.S. I asked this also at MathOverFlow.
The corresponding question on MathOverflow has been answered by darij grinberg in the negative: $I$ is not necessarily a biideal.