Clean coalgebras and clean comodules of finitely generated projective modules

Let $R$ be a commutative ring with multiplicative identity and $P$ is a finitely generated projective $R$-module. If $P^{\ast}$ is the set of $R$-module homomorphism from $P$ to $R$, then the tensor product $P^{\ast}\otimes_{R}P$ can be considered as an $R$-coalgebra. Furthermore, $P$ and $P^{\ast}$ is a comodule over coalgebra $P^{\ast}\otimes_{R}P$. Using the Morita context, this paper give sufficient conditions of clean coalgebra $P^{\ast}\otimes_{R}P$ and clean $P^{\ast}\otimes_{R}P$-comodule $P$ and $P^{\ast}$. These sufficient conditions are determined by the conditions of module $P$ and ring $R$.