Coideal
= Coideal
{title2=$\Delta I\subseteq I\otimes C+C\otimes I,\quad\varepsilon(I)=0$}
A <linear subspace> $I$ of a <coalgebra> satisfying the displayed conditions. These are exactly the conditions for the <comultiplication> and <counit> to descend to the quotient <vector space> $C/I$. An ideal that is also a coideal in a <bialgebra> gives a quotient <bialgebra>.