On Semiannahilator Supplement Submodules

Let R be associative; ring; with an identity and let D be unitary left R- module; . In this work we present semiannihilator; supplement submodule as a generalization of R-a- supplement submodule, Let U and V be submodules of an R-module D if D=U+V and whenever Y≤ V and D=U+Y, then annY≪R;. We also introduce the the concept of semiannihilator -supplemented ;modules and semiannihilator weak; supplemented modules, and we give some basic properties of this conseptes.

weakly supplement extending modules

In this paper, the extending property of modules is generalized by using weakly supplement submodules. We call a module M is weakly supplement extending if each submodule of M is essential in a weakly supplement submodule of M. Many characterization of weakly supplement extending module are obtained, we show that M is weakly supplement extending if and only if each closed submodule is weakly supplementing submodule of M. Moreover, we study the relation of weakly supplement extending module and among other known classes of the module such as lifting module, weakly supplemented module, supplement extending module and others. Also, we study conditions under it a direct sum of weakly supplement extending module is weakly supplement extending.

On S*-Supplemented Modules

The main goal of this paper is to introduce and study a new concept named d*-supplemented which can be considered as a generalization of W- supplemented modules and d-hollow module. Also, we introduce a d*-supplement submodule. Many relationships of d*-supplemented modules are studied. Especially, we give characterizations of d*-supplemented modules and relationship between this kind of modules and other kind modules for example every d-hollow (d-local) module is d*-supplemented and by an example we show that the converse is not true.

Strongly Weakly Supplement Extending Modules

In this paper, a class of modules which are proper strong concept of weakly supplement extending modules will be introduced and studied. We call a module M is strongly weakly supplement extending, if each submodule of M is essential in fully invariant weakly supplement submodule in M. Many characterizations of strongly weakly supplement extending modules are obtained. We show that M is strongly weakly supplement extending module if and only if every closed submodule of M is fully invariant weakly supplement submodule in M. Also we study the relation among this concept and other known concepts of modules. Moreover, we give some conditions that of strongly weakly supplement extending modules is closed under direct sum property is strongly weakly supplement extending.

Modules with finite spanning dimension

It is well known that if M is a module with finite spanning dimension, then one can talk of Sd(K), the spanning dimension of K only when K is a supplement submodule in M. In this paper we extend this concept to general submodules and obtained some important results. We characterize the set of all supplement submodules of the module R/(x) over R where R is a Euclidean domain and x ∈ R. Moreover, it is proved that the number of distinct supplements in R/(x) is 2k and Sd(R/(x)) = k where k is the number of distinct nonassociate prime factors of x.