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 $En{d}_R(M)$) is clean. Thus, any element of $En{d}_R(M)$ can be expressed as a sum of a unit and an idempotent element of $En{d}_R(M)$. Moreover, for a right $C$-comodule $M$, the endomorphism set of $C$-comodule $M$ denoted by $En{d}^C(M)$ is a subring of $En{d}_R(M)$. A $C$-comodule $M$ is a clean comodule if the $En{d}^C(M)$ 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.