The Necessary and Sufficient Condition of Clean R-Coalgebra e⇀C↼e

Let R be a commutative ring with multiplicative identity and C coassociative counital R-coalgebra the α-condition. A clean comodules defined based on cleanness rings modules. C-comodule M is comodule if endomorphism of clean. over itself i.e., as For an idempotent e ∈ R, there are relations between eRe R. It’s motivated us to investigate this condition for coalgebra. any C, we can construct ⇀C↼e where element dual algebra C. Here, show that conditions implies property ⇀C↼ vice versa.