On Maximal P-Injective Modules And Maximal P-Flat Modules

In the first section of this paper,we generalize P-injective modules and P-flat modules into maximal P-injective modules and P-flat maximal modules.Firstly,the definitions and some basic properties of maximal P-injective modules and maximal P-flat modules are discussed.Then we discuss the special properties of maximal P-injective modules and maximal P-flat modules on some special rings.The main results are obtained as follows:If R is a domain ring,then the sum of maximal P-injective modules is still maximal P-injective modules.If R is a left coherent ring,then right R-module M~is a maximal P-flat module if and only if M is a maximal P-injective module.The simple module on exchange ring is a maximal P-flat module if and only if it is a maximal P-injective module.In the second section,we introduce left maximal P hereditary ring, left maximal P coherent ring and maximal P regular ring.Moreover, these rings are characterized by maximal P-injective modules and maximal P-flat modules.At the same time,we gain an equational condition of maximal P regular ring by endomorphism ring:R is a maximal P regular ring if and only if when M_R is a maximal P-flat module andα∈End_RM,M/Imαis a maximal P-flat module if and only if when _RM is a maximal P-injective module andα∈End_RM,Kerαis a maximal P-injective module.In the third section,we introduce the concepts of maximal P-injective dimension and maximal P-flat dimension.Drawing on injective dimen- sion and flat dimension can be characterized by Ext and Tor,a new ring -mQP ring is introduced.Furthermore,we prove maximal P-injective dimension and maximal P-flat dimension also can be characterized by Ext and Tor if R is mQP ring.Then some properties and results of maximal P-injective dimension and maximal P-flat dimension on mQP ring are given.