Throughout this paper R represents commutative ring with identity and M is a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of weak essential submodules which introduced by Muna A. Ahmed, where a submodule N of an R-module M is called weak essential, if N ? P ? (0) for each nonzero semiprime submodule P of M. In this paper we rewrite this definition in another formula. Some new definitions are introduced and various properties of weak essential submodules are considered.
A gamma T_ pure sub-module also the intersection property for gamma T_pure sub-modules have been studied in this action. Different descriptions and discuss some ownership, as Γ-module Z owns the TΓ_pure intersection property if and only if (J2 ΓK ∩ J^2 ΓF)=J^2 Γ(K ∩ F) for each Γ-ideal J and for all TΓ_pure K, and F in Z Q/P is TΓ_pure sub-module in Z/P, if P in Q.
Let R be a commutative ring with identity, and let M be a unitary R-module. We introduce a concept of almost bounded submodules as follows: A submodule N of an R-module M is called an almost bounded submodule if there exists xÃŽM, xÃN such that annR(N)=annR(x).
In this paper, some properties of almost bounded submodules are given. Also, various basic results about almost bounded submodules are considered.
Moreover, some relations between almost bounded submodules and other types of modules are considered.
Let R be a commutative ring with unity and let M be a left R-module. We define a proper submodule N of M to be a weakly prime if whenever r  R, x  M, 0  r x  N implies x  N or r  (N:M). In fact this concept is a generalization of the concept weakly prime ideal, where a proper ideal P of R is called a weakly prime, if for all a, b  R, 0  a b  P implies a  P or b  P. Various properties of weakly prime submodules are considered.
Let R be commutative Ring , and let T be unitary left .In this paper ,WAPP-quasi prime submodules are introduced as new generalization of Weakly quasi prime submodules , where proper submodule C of an R-module T is called WAPP –quasi prime submodule of T, if whenever 0≠rstϵC, for r, s ϵR , t ϵT, implies that either r tϵ C +soc or s tϵC +soc .Many examples of characterizations and basic properties are given . Furthermore several characterizations of WAPP-quasi prime submodules in the class of multiplication modules are established.
Let R be a commutative ring with identity and M be an unitary R-module. Let ï¤(M) be the set of all submodules of M, and ï¹: ï¤(M)  ï¤(M)  {ï¦} be a function. We say that a proper submodule P of M is ï¹-prime if for each r  R and x  M, if rx  P, then either x  P + ï¹(P) or r M ïƒ P + ï¹(P) . Some of the properties of this concept will be investigated. Some characterizations of ï¹-prime submodules will be given, and we show that under some assumptions prime submodules and ï¹-prime submodules are coincide.
The main objective of this paper is to find the order and its exponent, the general form of all conjugacy classes, Artin characters table and Artin exponent for the group of lower unitriangular matrices L(3,? p ), where p is prime number.
The main purpose of this paper is to study feebly open and feebly closed mappings and we proved several results about that by using some concepts of topological feebly open and feebly closed sets , semi open (- closed ) set , gs-(sg-) closed set and composition of mappings.
Our aim in this paper is to introduce the notation of nearly primary-2-absorbing submodule as generalization of 2-absorbing submodule where a proper submodule of an -module is called nearly primary-2-absorbing submodule if whenever , for , , , implies that either or or . We got many basic, properties, examples and characterizations of this concept. Furthermore, characterizations of nearly primary-2-absorbing submodules in some classes of modules were inserted. Moreover, the behavior of nearly primary-2-absorbing submodule under -epimorphism was studied.
The significance of the work is to introduce the new class of open sets, which is said Ǥ- -open set with some of properties. Then clarify how to calculate the boundary area for these sets using the upper and lower approximation and obtain the best accuracy.