Exercise 23, § 11

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

(a) If C is coassociative (resp. cocommutative), so is every subcogebra. If $\gamma$ is a counit of C, the restriction of $\gamma$ to any subcogebra of C is a counit.

(b) Which are the subcogebras of the cogebra $A^{(X)}$ (no. 1, Example 4)?

(c) If $(S_\alpha)$ is a family of subcogebras of C, $\sum_\alpha S_\alpha$ is a subcogebra of C (the smallest containing all the $S_\alpha$).