Example

(5) It is immediate that the cogebra A (no. 1, Example 1) and the cogebra $A^{(X)}$ (II, § 11, no. 1, Example 4) are cocommutative. It follows from formula (5) of no. 1 that the cogebra $S(M)$ is cocommutative. Finally, for an A-algebra B such that the A-module B is projective and finitely generated to have the property that the cogebra $B^*$ (no. 1, Example 3) is cocommutative, it is necessary and sufficient that B be commutative; for (using the canonical identification of the A-module B with its bidual (II, § 2, no. 7)), this follows from the fact that the commutativity of diagram (14) is equivalent by transposition to that of the diagram which expressed the commutativity of B (§ 1, no. 3).