Abstract In previous research, a concept was presented f –small submodules A submodule L of an R-module W is named if L+X=W and W/X are small-singular, then X=W. this work, we present new concept, namely –hollow modules, -hollow every in -small submodule. This expansion Hollow modules. Some characteristics given to their relationship few the concepts comparability between them examined.

The purpose of this paper is to introduce new concepts in a module  over ring , the first one called e*- small essential submodule, which generalization second concept called e*-radical submodule radical and last e* - hollow module. We will prove some properties all these concepts.

the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d'subseteq m$ such that $n+d'$ is a strongly dense submodule of $m$...

All rings are commutative with 1 6= 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a, b ∈ R and m ∈M and abm ∈ N , then am ∈M -rad(N) or bm ∈M -rad(N) or ...

A new bidirectional isolated Zero-VoltageSwitching (ZVS) DC-DC converter with nonpulsating input and output current is proposed. This converter is ideally suited for battery charging application and is applied as a submodule in a battery storage system based on a modular multilevel converter (MC). The current output of the proposed DC-DC converter enables small current ripple for directly inter...

The modular multilevel converter (MMC) is receiving extensive research interests in high/medium voltage applications due to its modularity, scalability, reliability, high voltage capability and excellent harmonic performance. The submodule capacitors are usually quite bulky since they have to withstand fundamental frequency voltage fluctuations. To reduce the capacitance of these capacitors, th...

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

The purpose of this paper is to investigate pure submodules of multiplication modules. We introduce the concept of idempotent submodule generalizing idempotent ideal. We show that a submodule of a multiplication module with pure annihilator is pure if and only if it is multiplication and idempotent. Various properties and characterizations of pure submodules of multiplication modules are consid...

The purpose of this paper is to introduce a new concept in module M over ring R, called e∗-essential submodule, which generalization an essential submodule. We will some examples and properties about such that, what the inverse image intersection submodules direct sum submodules. show relationship between submodule Noetherian R-module. Also we define e∗-closed with