Intersection of coideals "Let $C$ be a coalgebra. If $I$ and $J$ are two coideals of $C$, show that $I\cap J$ is a coideal of $C$."
This is an exercise on page 45 of the book "Hopf Algebra" by Sweedler, Moss E(1969). But I doubt whether it is correct. The orthogonal of a coideal in the dual algebra is a subalgebra, and the sum of two subalgebra is not necessarily a subalgebra.
 A: I think this is a wrong proposition. Sweedler may make a mistake.
By the definition of I,J, $\epsilon(I \bigcap J) = 0$. $\triangle(I \bigcap J) \subset (I\otimes C + C\otimes I) \bigcap (J\otimes C + C\otimes J) $. Choose a linear basis, $\exists$ linear subspaces $W_I,W_J,V_I,V_J, I = (I \bigcap J) \oplus W_I , J = (I \bigcap J) \oplus W_J, C = I \oplus V_I, C = J \oplus V_J $.  Note that $W_I \bigcap W_J = 0, W_I \subset V_J$.
Then
$(I\otimes C + C\otimes I) \bigcap (J\otimes C + C\otimes J) = (((I \bigcap J) \oplus W_I) \otimes ((I \bigcap J) \oplus W_J \oplus V_J) + ((I \bigcap J) \oplus W_J \oplus V_J)\otimes ((I \bigcap J) \oplus W_I) ) \bigcap $
$(((I \bigcap J) \oplus W_J)\otimes ((I \bigcap J) \oplus W_J \oplus V_J) + ((I \bigcap J) \oplus W_J \oplus V_J)\otimes ((I \bigcap J) \oplus W_J) )$
It's easy to calculate intersections of direct summands. So
$(I\otimes C + C\otimes I) \bigcap (J\otimes C + C\otimes J) = (I \bigcap J)\otimes C + C \otimes (I \bigcap J) + W_I \otimes  W_J + W_J \otimes W_I$.
