Throughout this paper we introduce the notion of coextending module as a dual of the class of extending modules. Various properties of this class of modules are given, and some relationships between these modules and other related modules are introduced.

Let R be a commutative ring with identity, and let M be a unitary (left) R- modul e. The ideal annRM  = {r E R;rm  = 0 V  mE M} plays a central

role  in  our  work.  In  fact,  we  shall  be  concemed   with  the  case  where annR1i1 = annR(x) for   some   x EM such  modules  will  be  called bounded  modules.[t  htrns out that there are many classes of modules properly contained in the class of bounded modules such as cyclic modules, torsion -G·ee modulcs,faithful  multiplicat

Let  be a commutative ring with 1 and  be left unitary  . In this paper we introduced and studied concept of semi-small compressible module (a     is said to be semi-small compressible module if  can be embedded in every nonzero semi-small submodule of . Equivalently,  is  semi-small compressible module if there exists a monomorphism  , ,     is said to be semi-small retractable module if  , for every non-zero  semi-small sub module in . Equivalently,  is semi-small retractable if there exists a homomorphism  whenever  .

    In this paper we introduce and study the concept of semi-small compressible and semi-small retractable s as a generalization of compressible  and retractable  respectively and give some of

Let  be a commutative ring with 1 and  be left unitary  . In this papers we introduced and studied concept P-small compressible  (An     is said to be P-small compressible if  can be embedded in every of it is nonzero P-small submodule of . Equivalently,  is P-small compressible if there exists a monomorphism  , ,     is said to be P-small retractable if  , for every non-zero P-small submodule of . Equivalently,  is P-small retractable if there exists a homomorphism  whenever  as a generalization of compressible  and retractable  respectively and give some of their advantages characterizations and examples.

Let  be a commutative ring with 1 and  be left unitary  . In this paper we introduced and studied concept of semi-small compressible module (a     is said to be semi-small compressible module if  can be embedded in every nonzero semi-small submodule of . Equivalently,  is  semi-small compressible module if there exists a monomorphism  , ,     is said to be semi-small retractable module if  , for every non-zero  semi-small sub module in . Equivalently,  is semi-small retractable if there exists a homomorphism  whenever  .     In this paper we introduce and study the concept of semi-small compressible and semi-small retractable s as a generalization of compressible  and retractable  respectively and give some of their adv

      In this paper, we introduced module that satisfying strongly -condition modules and strongly -type modules as generalizations of t-extending. A module  is said strongly -condition if for every submodule of  has a complement which is fully invariant direct summand. A module   is said to be strongly -type modules if every t-closed submodule has a complement which is a fully invariant direct summand. Many characterizations for modules with strongly -condition for strongly -type module are given. Also many connections between these types of module and some related types of modules are investigated.

In this paper, a procedure to establish the different performance measures in terms of crisp value is proposed for two classes of arrivals and multiple channel queueing models, where both arrival and service rate are fuzzy numbers. The main idea is to convert the arrival rates and service rates under fuzzy queues into crisp queues by using graded mean integration approach, which can be represented as median rule number. Hence, we apply the crisp values obtained to establish the performance measure of conventional multiple queueing models. This procedure has shown its effectiveness when incorporated with many types of membership functions in solving queuing problems. Two numerical illustrations are presented to determine the validity of the

      The objective of this research is employ the special cases of  function  trapezoid in the composition of fuzzy sets to make decision within the framework of the theory of games traditional to determine the best strategy for the mobile phone networks in the province of  Baghdad and Basra, has been the adoption of different periods of the  functions belonging to see the change happening in the matrix matches and the impact  that the strategies  and decision-making  available to each player and the impact on  societ

Modeling data acquisition systems (DASs) can support the vehicle industry in the development and design of sophisticated driver assistance systems. Modeling DASs on the basis of multiple criteria is considered as a multicriteria decision-making (MCDM) problem. Although literature reviews have provided models for DASs, the issue of imprecise, unclear, and ambiguous information remains unresolved. Compared with existing MCDM methods, the robustness of the fuzzy decision by opinion score method II (FDOSM II) and fuzzy weighted with zero inconsistency II (FWZIC II) is demonstrated for modeling the DASs. However, these methods are implemented in an intuitionistic fuzzy set environment that restricts the ability of experts to provide mem