Exercise 24, § 11

Let C be a cogebra over a ring A. A sub-A-module S of C is called a right (resp. left) coideal if $c(S) \subset \operatorname{Im}(S \otimes C)$ (resp. $c(S) \subset \operatorname{Im}(C \otimes S)$).

(a) A subcogebra (Exercise 23) is a right and left coideal. The converse is true if A is a field.

(b) Every sum of right (resp. left) coideals is a right (resp. left) coideal.

(c) If S is a right (resp. left) coideal of C, the orthogonal $S^0$ is a right (resp. left) ideal of the dual algebra $C^*$.

(d) If A is a field and $\mathfrak{J}'$ a right (resp. left) ideal of the dual algebra $C^*$, the orthogonal $\mathfrak{{J}'}^0$ in C is a right (resp. left) coideal of C. (Argue by reductio ad absurdum by assuming that there exists an $x \in \mathfrak{{J}'}^0$ such that $c(x) \notin \mathfrak{{J}'}^0 \otimes C$; write $c(x) = \sum_i y_i \otimes z_i$, where the $z_i$ are linearly independent, and show that there would be $y' \in \mathfrak{J}'$ and $z' \in C^*$ such that $\langle y'z', x \rangle \neq 0.$)

Deduce that for a vector subspace S of C to be a right (resp. left) coideal, it is necessary and sufficient that $S^0$ be a right (resp. left) ideal of $C^*$.

(e) Deduce from (d) that if A is a field every intersection of right coideals (resp. left coideals, resp. subcogebras) of C is a right coideal (resp. left coideal, resp. subcogebra). For S to be a subcogebra of C, it is necessary and sufficient that its orthogonal S^0 be a two-sided ideal of the algebra C*.