Let $A$ be an algebra, $H$ a Hopf algebra, and 
$$
\beta\_A: A \to A \otimes H, ~~~~~ a \mapsto a^{(1)} \otimes a^{(2)}
$$
a right $H$-coaction. This induces a right $H$-coaction on $A \otimes A$ defined by
$$
\beta\_{A \otimes A}: a \otimes b \mapsto a^{(1)} \otimes b^{(1)} \otimes a^{(2)}b^{(2)}.
$$
My question is: Does this restrict to a coaction on the universal calculus over $A$, namely to a $H$-coaction on the kernel of the multiplication map $m:A \otimes A \to A$? I feel this is a very simple question but I can't seem to find an answer.


If the construction does not work, does anyone know of a way to induce a coaction on the universal calculus over $A$ from $\beta\_{A}$?



Coaction on the Universal Calculus 
If $A$ is an $H$-comodule algebra (that is, if the multiplication map $\mu$ is a map of comodules), then the answer is yes (trivially, because the category of $H$-comodules has kernels). If it isn't, then probably not, as then you have no compatibility between the algebra structure on $A$ and the comodule structure.


