| Injective modules and flat modules play very important roles in the studyof rings and categories of modules. In this thesis, we introduce the notion ofsmall injective module and small flat module, and study some properties aboutthem, and give some equivalent characterizations about them. We obtained aproposition that characterize J?semi-simple ring by small injective modules.In first chapter, it is necessary to preview some extensions about injectivemodules. Secondly, we introduce the notion of small injective module andstudy properties of small injective modules. in chapter 2, we introduce thenotion of small injective dimension, and obtain a proposition that characterizeJ-semi-simple rings by small injective dimensions, that is, r.small.id(R)=0 iffR is J-semi-simple ring. In chapter 3, we also introduce the notion of smallflat modules, and study equivalent characterizations and essential properties ofsmall flat modules. We discuss the relation between small flat modules andsmall injective modules, which generalizes the result of the Lambek rule. It isfrom the study of these that so much is learnt about small flat modules. |
