Hom-coalgebras A°and Hom-Modules

In this paper, we mainly study the construction and related problems of Hom-coalgebra. We also study the basic properties of Hom-coalgebra induced by Hom-algebra.The paper is mainly made up of three parts as follows:In the first section, we introduce the basic definitions and propositions used in this paper.In the second section, we discuss the relationships of Hom-ideals and Hom-coalgebras. We also study the Hom-coalgebras morphism. Furthermore, we define the Hom-algebras ideals, give the relationships of Hom-algebras ideals and Hom-coalgebras (Theorem2.1).In the third section, we define the cofinite two-sided ideals and study the related properties. In addition, we also obtain the sufficient conditions under which the Hom-coalgebras can be induced by hom-algebras in the general case (Theorem3.2). Further, we discuss the relationships of Hom-module and Hom-coalgebras morphism under the sufficient conditions mentioned above (Proposition3.4-3.9).At last, we give equivalent conditions with which one can judge wether an element belongs to the induced Hom-coalgebras (Theorem3.3).