On Pure Ideals: A Review

Abstract

Ring theory is one of the branches of an abstract algebra. This field is the study of a mathematical system with two binary operations. In this branch, many articles have studied this algebraic structure and presented some new works. However, the concept of purity has been studied before more than 40 years ago, especially the relation between the pure ideal and some other ideals on the given ring. In this paper, we survey the important results that concern with pure ideals. Some different types of ideals have been discussed such as N-pure ideals, z-ideals, Π-pure ideals and strongly pure ideals. Moreover, some recent results based on the work of several researchers have been summarized.  On the other hand, regarding these types of ideals, some questions have been presented. Furthermore, many important results about various types of rings which are based on the notion of pure ideals have been studied.