in this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.

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 $m$...

In a finite-dimensional vector space, every subspace is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. Unfortunately, the naive analogue of this for modules and submodules is wrong: (1) A submodule of a finitely generated module need not be finitely generated. (2) Even if a submodule of a finitely generated module is finitely generated, the minim...

The full-bridge submodules (FB-SMs) in hybrid modular multilevel converters need to absorb the enormous energy stored dc side and arm inductors during fault blocking, which may lead severe overvoltage. An energy-absorbing branch (EAB) composed of metal oxide varistor (MOV) thyristors is proposed this letter. No extra power loss produced by EAB normal operation. can a part reduce absorbed FB-SMs...