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 \(R\).