Annihilating coïdeal

Let $H$ be a $K$ -bimonoid and $𝑉, 𝑊$ be $H$ -modules. If $𝐼 \leq K H$ is a coïdeal such that $𝐼 𝑉 = 0$ and $𝑊 𝑉 = 0$ , then $𝐼 (𝑉 \otimes 𝑊) = 0$ .

Proof

Let $𝑥 \in 𝐼$ . Then $Δ (𝑥) = 𝑥 1 + 𝑥 2$ for some $𝑥 1 \in 𝐼 \otimes H$ and $𝑥 2 \in H \otimes 𝐼$ . Since $𝑥 1$ and $𝑥 2$ annihilate $𝑉 \otimes 𝑊$ , we have that $𝑥$ annihilates $𝑉 \otimes 𝑊$ .

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