Our active aim in this paper is to prove the following Let Ŕ be a ring having an
idempotent element e(e  0,e 1) . Suppose that R is a subring of Ŕ which
satisfies:
(i) eR  R and Re  R .
(ii) xR  0 implies x  0 .
(iii ) eRx  0 implies x  0( and hence Rx  0 implies x  0) .
(iv) exeR(1 e)  0 implies exe  0 .
If D is a derivable map of R satisfying D(R )  R ;i, j 1,2. ij ij Then D is
additive. This extend Daif's result to the case R need not contain any non-zero
idempotent element.

       Let R be a commutative ring with unity .M an R-Module. M is called coprime module     (dual notion of prime module) if ann M =ann M/N for every proper submodule N of M   In this paper we study coprime modules we give many basic properties of this concept. Also we give many characterization of it under certain of module.

  In this work we shall introduce the concept of weakly quasi-prime modules and give some properties of this type of modules.

      This study includedepidermal appendages characters of ten species belongingto the familyAsteraceae.Most of the studied species are characterized by eglandular, glandular, trichomesand papillae which are variable according to itskind, number of cells, shape, dimensions, characteristicsof surrounding cells, wall thickness and the position. Results showed that all species included glandular and eglandulartrichomes exceptCosmos bipinnatus, Ecliptaprostrate, andEupatoriumcannabinum that lacked in glandular trichomes and were based on their specific details.Trichomes were divided into different groups. The glandular ones involved sessile glandular or group multicellular and the head wasmace-shaped or round

    The main objective of this research is to study and to introduce a concept of strong fully stable Banach -algebra modules related to an ideal.. Some properties and characterizations of full stability are studied.

Let be a unitary left R-module on associative ring with identity. A submodule of is called -annihilator small if , where is a submodule of , implies that ann( )=0, where ann( ) indicates annihilator of in . In this paper, we introduce the concepts of -annihilator-coessential and - annihilator - coclosed submodules. We give many properties related with these types of submodules.

Let R be a commutative ring with identity and let M be a unital left Rmodule.
Goodearl introduced the following concept :A submodule A of an R –
module M is an y – closed submodule of M if is nonsingular.In this paper we
introduced an y – closed injective modules andchain condition on y – closed
submodules.

   Let R be a commutative ring with unity and let M be a unitary R-module. In this paper we study fully semiprime submodules and fully semiprime modules, where a proper fully invariant R-submodule W of M is called fully semiprime in M if whenever XXW for all fully invariant R-submodule X of M, implies XW.         M is called fully semiprime if (0) is a fully semiprime submodule of M. We give basic properties of these concepts. Also we study the relationships between fully semiprime submodules (modules) and other related submodules (modules) respectively.

In this thesis, we introduce eight types of topologies on a finite digraphs and state the implication between these topologies. Also we studied some pawlak's concepts and generalization rough set theory, we introduce a new types for approximation rough digraphs depending on supra open digraphs. In addition, we present two various standpoints to define generalized membership relations, and state the implication between it, to classify the digraphs and help for measure exactness and roughness of digraphs. On the other hand, we define several kinds of fuzzy digraphs. We also introduce a topological space, which is induced by reflexive graph and tolerance graphs, such that the graph may be infinite. Furthermore, we offered some properties of th