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Coideals of Hopf algebra coming from right (left) coideals K->K^+

If $H$ is a Hopf algebra over a field and $K$ is a 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 counter example.

Thanks!

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 Is $K^+$ defined as $K\cap \mathrm{Ker}\varepsilon$ or $\left(\eta\varepsilon-\mathrm{id}\right)\left(K\right)$ ? – darij grinberg Aug 8 2011 at 15:12