On Relative (m,n)-Projective,Injective And Flat Modules

Let m and n be non-negative integers and U a left R-module.In this article,we discuss the homological properties of(m,n)-projective,injective and flat modules relative to U.Firstly,we introduce the concepts of U-(m,n)-injective modules and U-(m,n)-flat modules,and discuss their properties.Secondly,U-(m,n)-projective modules are defined by U-(m,n)-injective modules and Ext functor,and its properties are characterized by the cotorsion pairs.Finally,the concept of strong(m,n)-coherent ring is introduced,the strong(m,n)-coherentness of endomorphism ring of general modules is discussed,and its properties are characterized by U-(m,n)-flat modules and U-(n,m)-presented modules.