Isolated Submodule And Its Generalizations

The isolated submodules play important roles in the study of rings and categories of modules.In the paper,some properties of isolated submodules are discussed. At the same time,definitions of strongly isolated submodules and locally isolated submodules are given and their properties are investigated.In chapter 1, we introduce the development of module theory and it's applications in the development of algebra,and present conditions of development of prime submodules,prime radicals,isolated submodules.Some definitions and results related to this topic are given in chapter 2.In chapter 3,some properties of strongly irreducible submodules and the relationships among strongly irreducible submodules,prime submodules, isolated submodules are considered.It is proven that the sum of two isolated submodules is still isolated.Finally,we discuss the properties of isolated submodules of fractional modules and the question when the homomorphic image of an isolated submodule is isolated.In chapter 4,the definition of strongly isolated submodules and their properties are introduced.It is shown that a supplemented submodule N of module M is strongly isolated if and only if there is a maximum submodule K of N for every proper submodule H of N such that H(?)K.This result generalizes[1, Proposition 1.4].In chapter 5,the notion of locally isolated submodules is posed and its properties are presented.