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 M is called St-closed in M, if N has no proper semi-essential extension in M, i.e if there exists a submodule K of M such that N is a semi-essential submodule of K, then N = K. We investigate the main properties of this type of submodules, and discuss some results that are useful in our work. The class of semi-extending modules is a generalization to the notion of extending modules, where an R-module M is called semi-extending, if every submodule of M is a semi-essential in a direct summand of M. Various properties of semi-extending modules are obtained, and we study the relationships between this class of modules and other related concepts.
In this paper, we introduce and study the concept of a new class of generalized closed set which is called generalized b*-closed set in topological spaces ( briefly .g b*-closed) we study also. some of its basic properties and investigate the relations between the associated topology.
The concept of closed quasi principally injective acts over monoids is introduced ,which signifies a generalization for the quasi principally injective as well as for the closed quasi injective acts. Characterization of this concept is intended to show the behavior of a closed quasi principally injective property. At the same time, some properties of closed quasi principally injective acts are examined in terms of their endomorphism monoid. Also, the characterization of a closed self-principally injective monoid is given in terms of its annihilator. The relationship between the following concepts is also studied; closed quasi principally injective acts over monoids, Hopfian, co Hopfian, and directly finite property. Ultimately, based on
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In this paper, we introduce the concepts of Large-lifting and Large-supplemented modules as a generalization of lifting and supplemented modules. We also give some results and properties of this new kind of modules.
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Abstract In this work we introduce the concept of approximately regular ring as generalizations of regular ring, and the sense of a Z- approximately regular module as generalizations of Z- regular module. We give many result about this concept.
A submodule Ϝ of an R-module Ε is called small in Ε if whenever , for some submodule W of Ε , implies . In this paper , we introduce the notion of Ζ-small submodule , where a proper submodule Ϝ of an R-module Ε is said to be Ζ-small in Ε if , such that , then , where is the second singular submodule of Ε . We give some properties of Ζ-small submodules . Moreover , by using this concept , we generalize the notions of hollow modules , supplement submodules, and supplemented modules into Ζ-hollow modules, Ζ-supplement submodules, and Ζ-supplemented modules. We study these concepts and provide some of their relations .
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The main goal of this paper is to give a new generalizations for two important classes in the category of modules, namely the class of small submodules and the class of hollow modules. They are purely small submodules and purely hollow modules respectively. Various properties of these classes of modules are investigated. The relationship between purely small submodules and P-small submodules which is introduced by Hadi and Ibrahim, is studied. Moreover, another characterization of purely hollow modules is considered.
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In this paper, we introduce the concepts of Large-lifting and Large-supplemented modules as a generalization of lifting and supplemented modules. We also give some results and properties of this new kind of modules.
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In this paper the full stable Banach gamma-algebra modules, fully stable Banach gamma-algebra modules relative to ideal are introduced. Some properties and characterizations of these classes of full stability are studied.
The present study involves experimental analysis of the modified Closed Wet Cooling Tower (CWCT) based on first and second law of thermodynamics, to gain a deeper knowledge in this important field of engineering in Iraq. For this purpose, a prototype of CWCT optimized by added packing under a heat exchanger was designed, manufactured and tested for cooling capacity of 9 kW. Experiments are conducted to explore the effects of various operational and conformational parameters on the towers thermal performance. In the test section, spray water temperature and both dry bulb temperature and relative humidity of air measured at intermediate points of the heat exchanger and packing. Exergy of water and air were calculated by applying the exergy
... Show MoreThis research deals with the concept of space in the theatrical performance and how the director works between two different spaces, the closed space and the open space. The question was how to use space according to the director's vision in the presentation. The problem of the research was whether the director of the Iraqi filmmaker could see the variable in the architectural space Or the place or space within the exhibition between open and closed, through the formation of the vision of the external components of intellectual and aesthetic and aesthetic, and impact on the relationship between the actor and the recipient to produce a new aesthetic space, and then the purpose of research and importance and limits and terminology The seco
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