Clean coalgebras and clean comodules of finitely generated projective modules. (English) Zbl 1501.16028
Summary: 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 \(P^{\ast}\otimes_RP\) can be considered as an \(R\)-coalgebra. Furthermore, \(P\) and \(P^{\ast}\) is a comodule over coalgebra \(P^{\ast}\otimes_RP\). Using the Morita context, this paper give sufficient conditions of clean coalgebra \(P^{\ast}\otimes_RP\) and clean \(P^{\ast}\otimes_RP\)-comodule \(P\) and \(P^{\ast}\). These sufficient conditions are determined by the conditions of module \(P\) and ring \(R\).
MSC:
| 16T15 | Coalgebras and comodules; corings |
| 16D90 | Module categories in associative algebras |
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
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