A module M over a ring R is said to be purely Rickart if the right annihilator in M of each endomorphism ring of a module M is a pure submodule of M. Purely Rickart module is a proper generalization of Rickart module. Some properties of the purely Rickart module are investigated. Also, we prove that the ring nxn matrix over R is a purely Rickart ring if and only if R is a weakly n-semiherditary ring. Every n-generated projective module is purely Rickart if and only if the free R-module R (n) is a purely Rickart. Others results are provided in this paper.

The main goal of this paper is introducing and studying a new concept, which is named H-essential submodules, and we use it to construct another concept called Homessential modules. Several fundamental properties of these concepts are investigated, and other characterizations for each one of them is given. Moreover, many relationships of Homessential modules with other related concepts are studied such as Quasi-Dedekind, Uniform, Prime and Extending modules.

In this paper normal self-injective hyperrings are introduced and studied. Some new relations between this concept and essential hyperideal, dense hyperideal, and divisible hyperring are studied.

A * -ring A is said to be strongly Rickart * -ring if x r

Let be a ring. Given two positive integers and , an module is said to be -presented, if there is an exact sequence of -modules with is -generated. A submodule of a right -module is said to be -pure in , if for every -Presented left -module the canonical map is a monomorphism. An -module has the -pure intersection property if the intersection of any two -pure submodules is again -pure. In this paper we give some characterizations, theorems and properties of modules with the -pure intersection property.