Let $k$ be a field, and let $S$ be a nonempty set. Let $k[S]$ be the grouplike coalgebra of $S$ over $k$, i.e. the free vector space with basis $S$ equipped with the coproduct $\Delta(s)=s\otimes s$ for all $s\in S$. (If $S$ is a group, then $k[S]$ is the usual group algebra of $S$ - which is also a Hopf algebra.) Let $I$ be a coideal in $k[S]$. Then, the coideal $I$ is spanned by differences $s-s'$ for some $s,s'$ in $S$.
This is Exercise 2.1.26(b) in Radford's Hopf algebras. It is clear that differences $s-s'$ for some $s,s'$ in $S$ always span a coideal in $k[S]$. But how to prove the other direction, that every coideal in $k[S]$ is of this form?
It should be simple because the other exercises in this book are not too hard. I was thinking that for a coideal $I$ in $k[S]$, we have $I\subseteq\bigoplus_{\text{for some }s\neq s'}k(s-s')$, where each summand $k(s-s')$ is a simple coideal, and conclude from this...
Another thought: As in other exercises in this book, I think the rank of $\Delta(c)$ where $c\in I$ plays a role, possibly of $c$ written as a sum with the least possible amount of nonzero summands $\lambda_{s,s'}(s-s')$ for some $s\neq s'$ ($\lambda_{s,s'}\in k$).