On Over Galois Rings Induced By Double-comodule Morita Context

Let H be a Hopf-algebra, A, B are right H comodule algebras. In [1], S. Caenepeel, S. Crivei, A. Marcus and M. Takeuchi. define H-Morita context, introduce the propertiesof Hopf-Galois extension, show the relations between the category of H-Morita context and the category of Morita context, and obtain the important theory: Let A, B be faithfully flat H-Galois extensions. If (A, B, M, N,α,β) is H-Morita context, then(A^coH,B^coH,M^coH,N^coH,α₁,β₁)is also Morita context. Conversely, if (A^coH,B^coH,M₁,N₁,α₁,β₁)is Morita context, then it can induce a H-Morita context.This paper wants to show that Morita-Takeuchi context of bicomodules over Galois coalgebras, and Morita context of bimodules over Galois corings. We obtain the main results as follows:In section 3, We consider the functors between the categories ^CM_H and ^CHM, define H-Morita -Takeuchi context. We obtain the similar results. It show the relations between the category of H-Morita- Takeuchi context and the category of Morita-Takeuchi context.In section 4, let (C,x) be At-coring, (D,y) be B-coring, we consider the adjoint functors(F,G), where F :A^coCM_B^coDâ†'^cM^D,F(M)=A(?)_A^coC M(?)_B^coDB;G:^CM^Dâ†'A^coCM_B^coD,G(N)=^coCN^coD,we obtain some results and define C-D-Morita context. Two importantresults of this section are: if(A,B,^CM^D,^DN^C,α,β)is C-D-Morita context, then(A^coC,B^coD,^coCM^coD,^codN^coC,α₁,β₁)is Morita context; conversely, if(A^coC,B^coD,M₁,N₁,α₁,β₁)is Morita context, then it can induce C-D-Morita context(A,B,A(?)_A^coC M(?)_B^coDB,B(?)_B^coDN(?)_A^coC A,α,β),these conclusions are the generalization of the results of [1].