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* is the set of R-module homomorphism from P to R, then the tensor product P* circle times P-R can be considered as an R-coalgebra. Furthermore, P and P* is a comodule over coalgebra P* circle times(R) P. Using the Morita context, this paper give sufficient conditions of clean coalgebra P* circle times(R) P and clean P* circle times(R) P-comodule P and P*. These sufficient conditions are determined by the conditions of module P and ring R.