Research On Modules And Self-injectiveness Of Formal Matrix Rings Of Order N

As a generalization of matrix rings,formal matrix rings are important research objects of algebra.Formal matrix rings of order n are developed from formal matrix rings of order 2.In 1958,Morita gave the notion of Morita Context and used it to study the equivalence of the category of modules.After 1973,AD.Sands and others began to study Morita context as a ring,called it Morita context ring,formal matrix ring,or formal matrix ring of order 2.The concept of formal matrix ring of order n was defined by?????????in 2003,but no substantive study on it was given.Since a formal matrix ring of order n can be seen as a formal matrix ring of order 2 through partition,and then it has not attracted much attentions until 2013.In 2013,Tang Gaohua and Zhou Yiqiang defined the formal matrix ring of order n over a ring R,and proved that it was determined by a set of central elements of R,denoted byM_n(R,s _ijk),which is a natural generalizition of the matrix ringM_n?R?.The formal matrix ringM_n(R,s _ijk)has abundant properties.Since then,several scholars begin to study the formal matrix rings of order n.But up to now,the research on the modules of the general formal matrix ring of order n and the self-injectivity of the general formal matrix ring?including the order 2?have not be found in literatures.In this paper,we mainly study the characterization of simple module,projective module,and injective module of general formal matrix rings and the self-injectivity and QF property of formal matrix rings.We have the following three chapters:The first chapter gives the research background of this article,and preliminary knowledge,and outlines the main results of this article.In chapter two,we discuss the basic properties of modules of general formal matrix rings,the construction of modules and module homomorphisms of general formal matrix rings.Superfluous and essential submodules,maximum and minimal submodules are discussed.Two special left modules:are given.By using of these two modules,projective and injective modules are characterized.In Chapter 3,we study the self-injectivity and QF property of general formal matrix ring by using the conclusions obtained in chapter 2.It is proved thatK_s?R?is a self-injective ring?QF ring?if and only if R is a self-injective ring?QF ring?.