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.

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 .

Abstract:

 Sought the search to find out what small businesses, the importance of funding, and the role of funders in supporting such projects, which have been addressed in the search through a sample of government banks (Rafidain and Rasheed) and private banks (the Middle East, the Gulf, North, Assyria) and a sample of Small projects borrowed from these banks, but in order to achieve that rely researchers on scientific methods discreet marked by the curriculum descriptive analytical method which is based on the compilation of facts and information then collated and analyzed to reach acceptable results, as well as data collection tool (checklist) was whereby examine questions of a sample of

Two field experiments were carried out for cultivating yellow maize crop Zea mays L. during the autumn planting season 2019 in two sites with soils of different textures. The first site is a loamy texture in one of the fields of the Medhatia Agriculture Division, Babylon Governorate. The second was silty loam by an alluvial mixture in one of the fields of Al-Nouriah Research Station, Ministry of Agriculture located in Al-Nouriah sub-district, Al-Qadisiyah governorate. It was found through the results that the uniformity, efficiency, and adequacy of the irrigation efficiency of the sprinkler irrigation method is better than that of the sprinkler irrigation method, and it ranged between (88.6-88.7) for uniformity and (84-86)% of the irrigatio

    In the presence of multi-collinearity problem, the parameter estimation method based on the ordinary least squares procedure is unsatisfactory. In 1970, Hoerl and Kennard insert analternative method labeled as estimator of ridge regression.

In such estimator, ridge parameter plays an important role in estimation. Various methods were proposed by many statisticians to select the biasing constant (ridge parameter). Another popular method that is used to deal with the multi-collinearity problem is the principal component method. In this paper,we employ the simulation technique to compare the performance of principal component estimator with some types of ordinary ridge regression estimators based on the value of t

Comes interest in the subject matter in the selection of the Family industry globally as one of the important industries which have developed a clear and significant in recent years, to look at these products and designs to their functional importance have appeared in recent years, the phenomenon of the small spaces because of housing Population density showed the need to find a spare Furniture fit these small spaces On this basis, determine the objective of this research is to arrive at a design techniques for dual-family employee for small spaces research sample included a double family manufactured in laboratories industry Furniture in Baghdad and local research sample includes family double Decker.Research focused on the first four c

In this work the interaction of plasma jet with water and hydrogen peroxide liquids used for assisted teeth bleaching by plasma jet had been study. A homemade plasma jet system was used. The plasma jet supply by 15 W electrical power generated by high voltage power supply of 9.6 kV peak to peak and frequency of 33 kHz .this power supply generate high electric field on electrodes that would be enough to ionize the argon gas. Some important agents were study such as the effect of the Ar gas flow rates on the length of the plasma jet, the influence of plasma jet on some properties of water and two hydrogen peroxide concentrations 25 % and 30 % like pH, conductivity and liquid temperature for different exposure time. The liquids temperature