@article{Puspita2022, 
author = {Nikken Prima Puspita and Indah Emilia Wijayanti},
title = {Bi-clean and clean Hopf modules},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {10},
pages = {18784-18792},
keywords = {clean ring, clean module, clean comodule, clean coalgebra, clean Hopf module, bi-clean Hopf module},
url = {https://www.sciopen.com/article/10.3934/math.20221033},
doi = {10.3934/math.20221033},
abstract = {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    E  n      d          R        (  M  )) is clean. Thus, any element of    E  n      d          R        (  M  ) can be expressed as a sum of a unit and an idempotent element of    E  n      d          R        (  M  ). Moreover, for a right    C-comodule    M, the endomorphism set of    C-comodule    M denoted by    E  n      d          C        (  M  ) is a subring of    E  n      d          R        (  M  ). A    C-comodule    M is a clean comodule if the    E  n      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.}
}