Let $A$ be a Hopf algebra (over a field). Consider a unital subalgebra $B\subseteq A$ with $\Delta(B)\subseteq B\otimes A$. Put
$$B^+:= B\cap \ker(\epsilon).$$
It can be shown that $B^+$ is a two-sided coideal of $A$, so that $AB^+$ is also a two-sided coideal.


**Question**: If $x\in AB^+$, is it true that $S(x) \in B^+ A?$


Of course, it suffices to show that $S(B^+)\subseteq B^+ A$ but I am not sure how to show this.


Thanks in advance for your help/comments/suggestions!



Hopf algebra and coideal question
This is indeed true, and it forms part of the proof of a result known as ***Koppinen's Lemma***.


The argument is as follows: Let $\{x\_j\,|\, j \in J\}$ be a basis of $B^+$, and take an element $x \in B^+$. It follows from the counit axiom of a Hopf algebra, and linear independence of our basis, that we can write the coproduct of $x$ as
$$
\Delta(x) = x \otimes 1 + \sum\_{j} x\_j \otimes y\_j, \textrm{ for some } y\_j \in A^+. 
$$
Applying $\epsilon \circ \eta = m \circ (S \otimes \mathrm{id})$ to this expression, we get the identity
$$
S(x) = - x\_jS(y\_j) \in B^+A.
$$


As I wrote above, this is a step in Koppinen's Lemma, which states, under the assumption of a bijective antipode, that
$$
S(AB^+) = B^+A. 
$$
An elementary proof (from which I took the above argument) along with a reference to the original article of Koppinin, can be found in this [article](https://link.springer.com/article/10.1007/BF02810683). (An open access copy can also be found on Schneider's LMU Munich website.)


It is also interesting to note that if $B^+A = AB^+$, then Koppinen's Lemma implies that the Hopf algebra structure of $A$ descends to the quotient $A/B^+A$.


