^{1}

^{1}

^{2}

^{1}

^{2}

Let

The motivation to introduce Hom-type algebras comes for examples related to

The crossed products of algebras were independently introduced in [

In [

This paper is organized as follows.

In Section

In this paper, all the vector spaces, tensor products, and homomorphisms are over a fixed field

We now recall from [

A Hom-algebra is a quadruple

A linear map

A Hom-coalgebra is a quadruple

A linear map

A Hom-bialgebra is a sextuple

Let

Let

Let

Let

Let

Let

Let

Recall from [

Let

Let

Let

If the cocycle

Directly computing, we can get that

Now, we prove that if

Conversely, suppose

If

For condition (C), the left hand side is

For condition

The automorphism

Now consider the coaction

We can prove that the condition (CU) and the condition (C) hold for any

(3) Let

If

From [

First, we prove that

Then we prove that

For a Hom-crossed coproduct, we can get the following properties, which are useful for the latter conclusions.

Let

(i)

(ii)

Applying

Note that if

The following result is the generalization of Proposition 2.1 in [

Let

Assume first that

Next we prove that if

Firstly, we prove that

Secondly, let

On the one hand,

Finally, we denote that

In this section, we introduce the notion of the cleft coextension and then discuss two equivalent characterizations of cleft coextensions.

Let

Let

By Proposition

The coextension

If

Conversely, if the coextension

Define

Recalling from [

Now we give a lemma, which will be used in the sequel.

Let

(1)

(2) If

(1) It is easy to see that

(2) In order to prove that

Then we discuss the relation between cleft coextension and Hom-module coalgebra with the Hom-Hopf module structure.

Let

Note that

(1) Now we prove that

(2) For all

Let

Assume that

Conversely, it is sufficient to show that

In this section, we always assume that

Now, we can obtain a Hom-coalgebra factorization for Hom-module coalgebra with the Hom-Hopf module structure.

Let

Since

We first verify that

Next we show that

By the above results, we obtain the following result.

Let

By Theorems

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the NNSF of China (no. 11601132), the Natural Science Foundation of Henan Province (no. 152300410086), the Research Fund of PhD of Henan Normal University (no. qd14151), the TianYuan Special Funds of the National Natural Science Foundation of China (no. 11626138), and the NSF of Shandong Province (no. ZR2016AQ03).