Bi-clean and clean Hopf modules

Let $ R $ be a commutative ring with multiplicative identity, $ C $ a coassociative and counital $ R $-coalgebra, $ B $ an $ R $-bialgebra. A clean comodule is a generalization and dualization of a clean module. An $ R $-module $ M $ is called a clean module if the endomorphism ring of $ M $ over $ R $ (denoted by $ End_{R}(M) $) is clean. Thus, any element of $ End_{R}(M) $ can be expressed as a sum of a unit and an idempotent element of $ End_{R}(M) $. Moreover, for a right $ C $-comodule $ M $, the endomorphism set of $ C $-comodule $ M $ denoted by $ End^{C}(M) $ is a subring of $ End_{R}(M) $. A $ C $-comodule $ M $ is a clean comodule if the $ End^{C}(M) $ is a clean ring. A Hopf module $ M $ over $ B $ is a $ B $-module and a $ B $-comodule that satisfies the compatible conditions. This paper considers the notions of a clean ring, clean module, clean coalgebra, and clean comodule in relation to the Hopf Module. We divide our discussion into two parts, i.e., clean and bi-clean Hopf modules. A $ B $-Hopf module $ M $ is said to be clean if the endomorphism ring of $ M $ is clean, and $ M $ is a bi-clean Hopf module if $ M $ is clean as a module over $ B $ and also clean as a comodule over $ B $. Moreover, we give sufficient conditions of (bi)-clean bialgebras and Hopf modules related to the cleanness concept of modules and comodules.