In this paper we generalize Jacobsons results by proving that any integer  in   is a square-free integer), belong to . All units of  are generated by the fundamental unit  having the forms

Our generalization build on using the conditions

This leads us to classify the real quadratic fields  into the sets  Jacobsons results shows that  and Sliwa confirm that  and  are the only real quadratic fields in .

The research discusses the most important goals and means of IGAD in making peace in Sudan and then regional peace.

The subject of multi- ethnics is one of the most important subjects in the study of political
geography, as multi- ethnics and its consequent problems are global geopolitical phenomena
that started early and reached its peak with the beginning of the twentieth century, because of
major changes in the political landscape that resulted by wars and led to the collapse of many
empires and major powers, a matter which led to put new political maps according to certain
considerations of the colonial powers, especially in Africa and Asia. All these things led to
the most serious challenges based on ethnic and sectarian conflict and led to the development
of geopolitical problems. Among the examples what most countries in th

In this paper we will study some of the properties of an operator by looking at the associated S-act of this operator, and conversely. We look at some operators, like one to one operators, onto operators. On the other hand, we look at some act theoretic concepts, like faithful acts, finitely generated acts, singular acts, separated acts, torsion free acts and noetherian acts. We try to determine what properties of T make the associated S-act has any of these properties.

Abstract

Toady technology Nano since it launched human Alivia third interest and anticipation unusual and that by the decision-makers in the world's governments and its institutions and research departments productivity, especially after the US Congress enact a law (the National Initiative for Tknlojba nanotechnology in the United States of America) and the allocation of large sums of money To support this initiative, the research aims to shed light on the concept of technology nanotechnology, which is technology atheist and the twentieth century, and reflect its economic importance and in order to reach the objectives of the research has been the division of research into several axes has been allocated

This work, in fact, is an English language teaching study (henceforth, ELT) which may provide the inspiration for much –needed research on issues such as testing students' level of achievement in articulatory phonetics at the university level. It attempts to answer two questions: 1)What is college students' level of achievement in articulatory phonetics? 2- Are there any statistical significant differences in the achievement of college students in articulatory phonetics between the production and recognition testing techniques? This research falls into three sections: Section one indicates the problem, the aims, the hypotheses, the procedures, the limits and the value of the study. Section two provides a general overview of the articul

Represents light a sovereign and importance in cinematography element of what possessed the physical characteristics make it a sharp image detail or opaque, or any other form as he wants the director, Valdu away from the main function, which emerged with the beginnings of cinema, any access to endanger the proper and good, the Alachtgalat dramatic and psychological are deepened performance lighting and functioned in adjacent with the shadows, all the two complement each other, which affects the image details or even the ability to express their personal depths or the social and psychological same place, and this prompted the researcher to determine the search in the following address: (employing lighting to express their personal duplica

This research tackles the play of Synge "Rider to the Sea" to find out whether the
theories of Leech (1983) and Brown and Levinson (1987) can be applied or not to this study.
The model of Leech (1983) consists of seven principles and for Brown and Levinson
(1987),it consists of two principles of politeness where one of them is positive and the other is
negative politeness.
It is hypothesed that:
1. There is a relation between the two theories.
2. Synge deals with these theories in such a way to depict his characters.
3. It is also hypothesed that every character deals with these principles in such a way to show
the events of the play.
The study concludes that there is a relation between the two theories in d