In this paper, we introduce and study the notions of fuzzy quotient module, fuzzy (simple, semisimple) module and fuzzy maximal submodule. Also, we give many basic properties about these notions.

      In this paper, we introduce and study the notation of approximaitly quasi-primary submodules of a unitary left -module  over a commutative ring  with identity. This concept is a generalization of prime and primary submodules, where a proper submodule  of an -module  is called an approximaitly quasi-primary (for short App-qp) submodule of , if , for , , implies that either  or , for some . Many basic properties, examples and characterizations of this concept are introduced.

Let R be a commutative ring with identity and let M be a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of semi-essential submodules which introduced by Ali S. Mijbass and Nada K. Abdullah, and we make simple changes to the definition relate with the zero submodule, so we say that a submodule N of an R-module M is called semi-essential, if whenever N ∩ P = (0), then P = (0) for each prime submodule P of M. Various properties of semi-essential submodules are considered.

    Let  be a non-zero right module over a ring  with identity. The weakly second submodules is studied in this paper. A non-zero submodule  of   is weakly second Submodule when  ,  where ,  and  is a submodule of  implies either  or   . Some connections between these modules and other related modules are investigated and number of conclusions  and characterizations are gained.

 Let  be a commutative ring with identity and a fixed ideal of  and  be an unitary -module.We say that a proper submodule  of  is -semi prime submodule if with . In this paper, we investigate some properties of this class of submodules. Also, some characterizations of -semiprime submodules will be given, and we show that under some assumptions -semiprime submodules and semiprime submodules are coincided.

Let  be a ring with identity and  be a submodule of a left - module . A submodule  of  is called - small in  denoted by , in case for any submodule  of ,  implies .  Submodule  of  is called semi -T- small in , denoted by , provided for submodule  of ,  implies that . We studied this concept which is a generalization of the small submodules and obtained some related results

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 we introduced many new concepts all of these concepts completely
depended on the concept of feebly open set. The main concepts which introduced in
this paper are minimal f-open and maximal f-open sets. Also new types of
topological spaces introduced which called Tf min and Tf max spaces. Besides,
we present a package of maps called: minimal f-continuous, maximal f-continuous,
f-irresolute minimal, f-irresolute maximal, minimal f-irresolute and maximal firresolute.
Additionally we investigated some fundamental properties of the concepts
which presented in this paper.

In this paper, we introduce and study the notion of the maximal ideal graph of a commutative ring with identity. Let R be a commutative ring with identity. The maximal ideal graph of R, denoted by MG(R), is the undirected graph with vertex set, the set of non-trivial ideals of R, where two vertices I₁ and I₂ are adjacent if I₁ I₂ and I₁+I₂ is a maximal ideal of R. We explore some of the properties and characterizations of the graph.

In''this"article, we"study",the"concept""of WN"-"2"-''Absorbing'''submodules and WNS''-''2''-''Absorbing"submodules as generalization of "weakly 2-absorbing and weakly semi 2-absorbing submodules respectively. We investigate some of basic properties, examples and characterizations of them. Also, prove, the class of WN-2-Absorbing "submodules is contained in the class of WNS-2-Absorbing "submodules. Moreover, many interesting results about these concepts, were proven.