Generalized Injectivity Of Rings And Modules And Its Applications

This dissertation, consisting of six chapters, is concerned with several gener-alized injectivity of rings and modules. Unless otherwise mentioned, in this thesis, R will denote an associative ring with identity 1≠0.Chapter 1 introduces the background and the main results obtained in this thesis, and lists the main concepts and terminologies.In Chapter 2, we introduce right PS-injective rings as a proper generaliza-tion of right P-injective rings and right small injective rings. We first give some examples to show that the class of right PS-injective rings is a proper subclass of right mininjective rings, and research some properties of right PS-injective rings. It is shown that PS-injective rings are not Morita invariant and PS-injective rings need not be two-side symmetric. Meanwhile, we prove that, if R is a semiregular ring, then R is right P-injective if and only if R is right PS-injective. Thus, many properties of right P-injective rings can be extended to right PS-injective rings. Next, in Section 3 of this chapter, the concept of left J-morphic rings is intro-duced as a generalization of left morphic rings. It is shown that J-morphic rings are not quasi-morphic. We prove that, if R is a local ring, then R is left morphic if and only if R is left J-morphic. Moreover, we show that left J-morphic rings are right PS-injective. So some properties of left morphic rings can be extended to left J-morphic rings. Section 4 considers right PS-injective rings under trivial extensions. It is proven that, if S=R∝R is right PS-injective, then R is right P-injective. Finially, as an application of PS-injective rings, we give some new characterizations of GPF-rings and QF-rings.Chapter 3 investigates the application of PS-injective modules. We introduce PS-flat modules as a dual concept (not in the sense of category). The relations be-tween PS-injective modules and PS-flat modules over spacial rings are researched. Using the two classes of modules, we give some characterizations of semiprimitive rings. Section 3 introduces the concept of left PS-coherent rings as a proper gen- eralization of left J-coherent rings and left P-coherent rings. It is shown that R is left J-coherent if and only if Mn(R) is left PS-coherent for all n≥1. Meanwhile, we prove that, if R is a semiregular ring, then R is left P-coherent if and only if R is left PS-coherent. We obtain the necessary and sufficient conditions that R is a left PS-coherent ring in terms of PS-injective modules and PS-flat modules. We also discuss the existence of PS-injective covers and PS-flat preenvelopes over PS-coherent rings.Chapter 4 is concerned with the FP-small injectivity and J-injectivity of rings and modules. As a proper generalization of FP-injectivity of rings and modules, we introduce FP-small injectivity, and show that FP-small injective rings are Morita invariant but need not be two-side symmetric. Some properties of FP-small injective rings similar to that of FP-injective rings are obtained. We have the following result:If R is a semiregular ring, then R is right FP-injective if and only if R is right FP-small injective. So we can characterize FP-rings and QF-rings in terms of FP-small injective rings. Following [32], Section 3 considers J-injectivity of rings and modules, and discusses the relations between J-injectivity and other generalized injectivity such as small injectivity,f-injectivity and PS-injectivity. In the end of this chapter, we characterize semiprimitive rings in terms of FP-injectivity and J-injectivity of modules.Chapter 5 studies FGT-injectivity of modules. In Section 2, we consider the relations between the FGT-injective dimension of rings (modules) and the existence of some special FGT-injective (pre)covers and FGT-flat (pre)envelopes of modules over rightΠ-coherent rings. It extends some results in [65], respectively. Section 3 researches whether FGT-injectivity, FGT-flatness andΠ-coherence are preserved under almost excellent extensions. It is shown that, if S is an almost excellent extension of a right II-coherent ring R, then S is rightΠ-coherent. Section 4 discusses the existence of TIn-covers and TFn-preenvelopes of modules over right II-coherent rings. In Section 5, we introduce the concepts of n-TI-injective modules and n-TI-flat modules, and study the relation between n-TI-injective modules and TIn-covers and the relation between n-TI-flat modules and TFn-preenvelopes, respectively. We prove that R is a QF-ring if and only if any left (right) R-module is n-TI-injective. We also give some characterizations of weak n-Gorenstein rings in terms of n-TI-injective modules and TIn-covers.Chapter 6 lists some problems that have not been resolved in this thesis.