On Strongly Superfluous Submodules And Semiregularity Of Modules

In this thesis, we use strongly superfluous submodules to introduce and study amply strongly-supplemented modules, strongly singular modules and strongly-(S*) modules, and obtain properties of strongly superfluous submodules in supplement-ed modules. We also investigate l-semiregularity and l-semiperfecty of rings and modules by l-superfluous submodules.There are three chapter in this thesis:In Chapter One, we use strongly superfluous submodules to define and character-ize amply strongly-supplemented modules, strongly singular modules and strongly-(S*) modules, we obtain some properties of strongly superfluous submodules in supplement modules, give sufficient conditions of the nilpotency of the strongly radical of modules. We also prove that strongly singular modules are closed under submodules, homomor-phic image and summand, give some properties of strongly-(S*) modules and some equivalent characterizations of semisimple module M/Z*8.8S(M).In Chapter Two, we use l-superfluous submodules to define l-semiregular ele-ments and give some equivalent characterizations of l-semiregular elements and l-semiregular modules. Under some conditions, we show that M is l-semiregular if and only if M/l(M) is regular. By the DM conditons we define DMl projective covers, and use them to characterize l-semiregularity of rings. Under Morita equivalence, we investigate the properties of l-semiregular modules.Basing on the former chapter, in Chapter Three, we define and study l-semiperfect modules and l-semiperfect rings. We prove that a finite direct sum of projective l-semiperfect module is l-semiperfect. We also use DMl projective covers to charac-terize l-semiperfect rings.