The Soc-quasi-Injective Modules

Injective modules play an important role in module theory and homo-logical algebras and have many generalized forms. In this thesis, we shall discuss the properties of soc-quasi-injective modules by some known proper-ties of quasi-injective modules and soc-injective modules. It consists of the three chapters.In chapter 1, we introduce injective modules, some generalized forms of injective modules, and our main work in this thesis.In chapter 2, we introduce the notions of soc- injective modules and soc-quasi-injective modules, and obtain some properties of soc-quasi-injective mod-ules. We shall prove that the class of soc-injective R-modules is closed under isomorphism, direct products, finite direct sums and direct summands. By a new way, we show that the soc-quasi-injective module satisfies the C2 and C3 conditions. Some results generalize the known properties of soc-quasi-injective modules. By the decomposition of the direct sum, we give the relation between the injective module and the soc-quasi-injective module. Finally, we focus on the soc-quasi-injective modules on some special rings.The third chapter introduce the notions and some related properties of soc-M-injective rings and soc-injective rings. Furthermore, we define a new homological dimension, say the soc-injective dimension, and prove that a right R-module N is a soc-M-injective module if and only if ExtR1(M/soc(M), N)= 0, and a right R-module N is soc-M-injective module with sid(N)< 1 if and only if there exists an epimorphism of modules h:N1→N2 such that N≌Kerh, where N1; N2 are soc-M-injective modules.