Let R be a commutative ring with identity 1 ¹ 0, and let M be a unitary left module over R. A submodule N of an R-module M is called essential, if whenever N ⋂ L = (0), then L = (0) for every submodule L of M. In this case, we write N ≤e M. An R-module M is called extending, if every submodule of M is an essential in a direct summand of M. A submodule N of an R-module M is called semi-essential (denoted by N ≤sem M), if N ∩ P ≠ (0) for each nonzero prime submodule P of M. The main purpose of this work is to determine and study two new concepts (up to our knowledge) which are St-closed submodules and semi-extending modules. St-closed submodules is contained properly in the class of closed submodules, where a submodule N of

This article is devoted to the cognitive study of ironic metonymy in Russian and Arabic. Metonymy and irony have traditionally been seen as parallel linguistic phenomena. But their formation and interpretation are based on different cognitive mechanisms. At the formal and functional level, metonymy and irony have a number of significant differences. Metonymy is an artistic technique, the mechanism of which is based on obvious, easily traced connections between objects and phenomena of the surrounding world. Irony is a satirical technique or a rhetorical figure that is used to create a certain artistic image, aimed at forming the hidden meaning of the statement. A native speaker intuitively feels the difference between metonymy and i

The current challenge facing our age is the information digitization, at the same time, there is a huge development in the interdisciplinary of technologies with sciences. In the last few years, the World witnessed a number of information challenges with different dimensions, including digital dimensions, which are called the digital world and the virtual world for authors and movie writers. Consequently, the digital art concept emerged that utilizes the computer in an efficient way and as a new technique for drawing. This art is considered a great leap for modern art. The influence of digital technology transformed traditional arts like painting and sculpture into new forms, from pure arts to virtual reality. The most important characte

      In this work we discuss the concept of pure-maximal denoted by (Pr-maximal) submodules as a generalization to the type of R- maximal submodule, where a proper submodule  of an R-module  is called Pr- maximal if  ,for any submodule  of W is a pure submodule of W, We offer some properties of a Pr-maximal submodules, and we give Definition of the concept, near-maximal, a proper submodule  

 of an R-module  is named near (N-maximal) whensoever  is pure submodule of  such that  then K=.Al so we offer the concept Pr-module, An R-module W is named Pr-module, if every proper submodule of  is Pr-maximal. A ring  is named Pr-ring if whole proper ideal of  is a Pr-maximal ideal, we offer the concept pure local (Pr-loc

Despite the history of Baghdad city extends into a long history, most of the contemporary buildings of Baghdad have been shaped in the era of modernity. Furthermore, most of the buildings of modernity in Baghdad are types of modernity buildings in Iraq as a whole, and due to all the joints of change and development are taking place in Iraq starts from Baghdad. Accordingly, all selected buildings, which would be presented as case studies of modernity will be exclusively in Baghdad. Although the importance of this significant modernist product, which represents the identity of Baghdad, which should be preserved by the renewal and preservation policies, the problem of research was emerged as follow: new fin

Let M be an R-module, where R be a commutative;ring with identity. In this paper, we defined a new kind of submodules, namely; ET-coessential and ET-Coclosed submodules of M. Let T be a submodule of M. Let K  H  M, K  is called  ET-Coessential of H in M (K⊆_ET.ce H), if     . A submodule H is called ET- coclosed in M of H has no proper coessential submodule in M, we denote by  (K⊆_ET.cc H) , that is, K⊆_ET.ce H implies that   K = H. In our work, we introduce;some properties of ET-coessential and ET-coclosed submodules of M.

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Isthmus life and prepare for it

   Let R be a ring with identity and Ą a left R-module. In this article, we introduce new generalizations of compressible and prime modules, namely s-compressible module and s-prime module. An R-module A is s-compressible if for any nonzero submodule B of A there exists a small f in Hom_R(A, B). An R-module A is s-prime if for any submodule B of A, ann_R (B) A is small in A. These concepts and related concepts are studied in as well as  many results consist properties and characterizations are obtained.