The Study On Some Properties Of Coring

For a long period, the theory of modules over rings on the one side and comodules and Hopf modules for coalgebras and bialgebras on the other side developed quite independently. When coring came up, this gap realised. Graded modules , Hopf modules, Long dimodules, Yetter-Drinfeld modules, entwined modules and weak entwined modules are special cases of comodules over a coring, so module theory can be applied to enrich the theory of comodules and vice versa. Coring can also be used to study properties of functors between categories of these modules. For all , Coring become an object that is being paied close attention to by algebra scholars.In 1975, Corings were introduced by M.Sweedler in[9], to give a formulation of a predual of the Jacobson-Bourbakis theorem for intermediate extensions of division ring extensions.. A coring over an associative ring with unit is a coalgebra in the monoidal category of all A-bimodules. In 1998, motivated by an observation of M.Takeuchi[10]namely that new example of corings can be provided by entwining structure. In 2002, T.Brzezinski in[20]discussed a series of papers illustrating the importance of coring. In[20], T.Brzezinski has given some new examples and general properties of corings additive Galois coring. In 2003, the notion of Galois corings was extened to comodules by L.EI Kaoutit and J.Gomez-Torrecillas in[7]. While a coalgebra can be understood as a dualisation of an algebra, a coring is a dualisation of ring. In this paper, we build and analyze the duality of a coring-C ring,C is a coalgebra. We generalise results in corings, and made some conclusions in C-rings.The thesis is composed of five chapters.Chapter One is the introduction of this paper, Which introduces the problem and the background of the problem we have studied.Chapter two is preparatory knowledge, Which contains four sections. We introduce the basic notions and conclusions of corings;Galois-type theory in corings , and a special class of corings termed comatrix-corings.In Chapter Three, first we introduce the dualisation of the notion of a coring which is a C-ring, study properties of C-ring and build several examples of a C-ring. Second we study the forgetful functor from the category of right modules of an C-ring -4-to the category of right C-comodules and the introduction functor —OCA of C-ring . Last, we study when these functors are separable and the properties of adjoint of these functors.In Chapter Four, first we introduce entwining structure^, C, ip) and entwining modules category. Build a C-ring A= C A, made the conclusion Ma= M(ip)%, at the same time we use the C-ring further study the relations between entwining module category and A-Galois coextension . Second , we introduce ring-ring weak entwining structure (A, C, V>?), build a C-ring A= ItuPr, find the conclusion Ma= M(V>h)S- At last , we build a corresponding C-ring B= IjuPl, on the basic of left-left weak entwining structure additive Pi, find out that Ms=cA M{$>i).In Chapter Five , on the basic of 2.4, we introduce the dualisation the notion of comatrix coring, named matrix C-ring. We study the relative functors of matrix C-ring , and Galois theory applied to matrix C-ring.