Is the coaction $\delta: A \to H \otimes A$ injective?

Question

Let $A$ be an algebra and let $H$ be a bialgebra. Suppose that $A$ is an $H$-comodule. Then we have a coaction $\delta: A \to H \otimes A$. Is the coaction $\delta: A \to H \otimes A$ always injective? Thank you very much.

Answer 1

Yes, it is even a split monomorphism: If $(C,\Delta,\eta)$ is a ${\mathbb k}$-coalgebra and $(M,\nabla)$ is a $C$-comodule, then (by definition) you have that $M\xrightarrow{\nabla} M\otimes_{\mathbb k} C\xrightarrow{\text{id}_M\otimes\eta} M\otimes_{\mathbb k} {\mathbb k}\cong M$ is the identity on $M$. In particular, $\nabla: M\to M\otimes_{\mathbb k} C$ is injective. As Tobias said, this is dual to the perhaps more obvious (by habit) argument that the action of any (unitary) ring on any of its modules is surjective: just look at the action of the unit.