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If $H$ is a Hopf algebra over a field and $K$ is a right or left coideal of $H$ then $K^+=K\cap\ker\epsilon$ is a coideal of $H$. Does this hold when $k$ is a commutative ring? If not, what is a counterexample?

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(Note: It is rather easy to prove that $\left(\eta \varepsilon - \operatorname{id}\right)\left(K\right)$ is a coideal of $H$, because every $k \in K$ satisfies $\Delta\left(k - \varepsilon\left(k\right)\right) = \sum_{(k)} \left(k_{(1)} - \varepsilon\left(k_{(1)}\right)\right) \otimes k_{(2)} + 1 \otimes \left(k - \varepsilon\left(k\right)\right)$, using Sweedler's notation.) –  darij grinberg Aug 1 at 12:47

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