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Let (H,α) be a Hom-Hopf algebra and (C,β) a Hom-coalgebra. In this paper, we first introduce the notions of Hom-crossed coproduct C ⋊λ H and cleft coextension and then discuss the equivalence between them. Furthermore, we discuss the relation between cleft coextension and Hom-module coalgebra with the Hom-Hopf module structure and obtain a Hom-coalgebra factorization for Hom-module coalgebra with the Hom-Hopf module structure.

The Factorization for a Class of Hom-Coalgebras