Simple Utumi Modules And Maximal Dual Utumi Modules

In chapter one,the development of the theory of rings and categories of modules and the present research status at home and abroad are introduced.In chapter two,some important related definitions,lemmas and conclusions needed in this paper are introduced.In chapter three,the concept of simple Utumi modules?simple U-modules for short?and singular simple Utumi modules?singular simple U-modules for short?are introduced.Their fundamental properties are studied and some classical classes of rings and modules are characterized.A right R-module M is called a simple U-module if for any two non-zero simple submodules A and B of M with A?31?B and A?B?28?0,there exist two direct summands K and L of M such thatA?^essK,B?^essL andK?L?^?M;a right R-module M is called a singular simple U-module if for any two non-zero singular simple submodules A and B of M with A?31?B and A?B?28?0,there exist two direct summands K and L of M such thatA?^essK,B?^essL andK?L?^?M.Some examples,which are simple U-modules but not U-modules and which are singular simple U-modules but not simple U-modules,are given.The equivalent conditions of simple-direct-injective modules are given.It is proven that simple U-modules are simple-direct-injective modules.The equivalent characterizations of singular simple-direct-injective modules are given.It is proven that singular simple U-modules are singular simple-direct-injective modules.Some examples are given to show that the direct sums of simple U-modules are usually not simple U-modules.In the mean time,some sufficient conditions when the direct summands of a simple U-module and direct sums of simple U-modules are simple U-modules are given.It is proven that a ring R is a right V-ring if and only if each right R-module is a simple U-module;a ring R is a right GV-ring if and only if each right R-module is a singular simple U-module;a ring R is semisimple Artinian ring if and only if each simple U-module is an injective module;a ring R is a regular right V-ring if and only if each cyclic right R-module is a simple U-module;a ring R is a right Notherian V-ring if and only if each simple U-module is a strongly Soc-injective module;a ring R is a regular right GV-ring if and only if each cyclic right R-module is a singular simple U-module.In chapter four,the concept of maximal Dual Utumi modules are introduced?maximal DU-modules for short?.Their fundamental properties are studied,the relationships between maximal DU-modules and simple U-modules are discussed,and some classical classes of rings and modules are characterized.Some examples which are maximal DU-modules but not DU-modules are given.The equivalent characterizations of simple-direct-projective modules are given.It is proven that maximal DU-modules are simple-direct-projective modules.Some examples are given to show that direct sums of maximal DU-modules are usually not maximal DU-modules,and some sufficient conditions when the factor modules of a maximal DU-module are maximal DU-modules are given.It is proven that a ring R is a semisimple ring if and only if each right R-module is a maximal DU-module;if each simple Utumi right R-module is a maximal Dual Utumi right R-module,then R is a semi-local ring.