Prime ideals and prime submodules

Prime ideals have an important role in commutative algebra. One of the well known properties of prime ideals is the following: If a prime ideal contains a finite intersection of a family of ideals then some of those ideals are contained in the prime ideal. Reviewing the literature, we have seen that authors discuss this property in the case of the infinite intersection in detail and that rings in which every prime ideal has this property in the infinite case are called strongly 0-dimensional rings. These rings are also the dual notion of compactly packed rings. When we examined the studies on the infinite union of prime ideals, we have seen the concept of coprimely packed rings which are a generalization of compactly packed rings. With the aid of the motivation gained by these studies, we define coprimely structured rings as follows: A prime ideal of a ring is said to be a coprimely structured ideal if, whenever it is coprime to each element of a family of ideals of the ring, it does not contain any intersection of ideals in this family. We say that a ring is coprimely structured if every prime ideal of it is coprimely structured. In this study, we work on some properties of coprimely structured rings and we examine localization of these rings. Furthermore, we investigate coprimely structured rings and give some relations between coprimely structured rings and other rings such as Artinian rings, strongly 0-dimensional rings, rings satisfying *-property. Moreover, we show that coprimely structured rings are a general expression of h-local domains and we examine under which conditions any h-local domain is a coprimely structured ring. As it is mentioned above; if a prime ideal contains a finite intersection of ideals then some of the ideals are contained in the prime ideal. This property does not valid for submodules in general. That is, if a prime submodule contains a finite intersection of submodules, it could be the case that none of the ideals is contained in the prime submodule. However, as in the case of commutative rings, multiplication modules satisfy this property. For the purpose of investigating this property with infinite intersection in multiplication modules, we define strongly 0-dimensional modules. This notion is a version of strongly 0-dimensional rings in module theory. Also, it is an extension of strongly 0-dimensional rings and it is defined as follows: A prime submodule of a multiplication module is called a strongly prime submodule if, whenever the prime submodule contains any intersection of submodules of the module, one of the submodules is contained in the prime submodule. A multiplication module is said to be a strongly 0-dimensional module if any prime submodule of it is a strongly prime submodule. We investigate properties of strongly 0-dimensional modules and give some relations between von Neumann regular modules, Q-modules and strongly 0-dimensional modules.