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 .

Background: The global threat of COVID-19 outbreak and on the 11 March 2020, WHO acknowledged that the virus would likely spread to all countries across the globe and declared the coronavirus outbreak a pandemic which is the fifth pandemic since 20 century and this has brought human lives to a sudden and complete lockdown and the confirmed cases of this disease and deaths continue to rise in spite of people around the world are taking important actions to mitigate and decrease transmission and save lives. Objectives: To assess the effect of exercise and physical activity on the immunity against COVID-19. Methods: Collected electronic databases including (Medline, EMBASE, Google Scholar, PubMed and Web of Science) were searched with

The research aims to identify the most important variables affecting shooting from jumping high and compare them for the two foot the weak and strong, where the researchers adopted the descriptive method, and the sample was chosen by the intentional method, which consists of (4) players from the Iraqi Sports Army Club, where these variables were studied and their impact on The accuracy of aiming at the two men, and the researchers concluded that most of the players have more accuracy in aiming at the strong leg than at the weak leg, which leads to the loss of many real opportunities during the match because of the players changing the situation or wasting the available opportunity when the position of correction is an opportunity for the w

Two Schiff bases, namely, 3-(benzylidene amino) -2-thioxo-6-methyl 2,5-dihydropyrimidine-4(3H)-one (LS])and 3-(benzylidene amino)-6-methyl pyrimidine 4(3H, 5H)-dione(LA)as chelating ligands), were used to prepare some complexes of Cr(III), La(III), and Ce(III)] ions. Standard physico-chemical procedures including metal analysis M%, element microanalysis (C.H.N.S) , magnetic susceptibility, conductometric measurements, FT-IR and UV-visible Spectra were used to identify Metal (III) complexes and  Schiff bases (LS) and (LA). According to findings, a [Cr(III) complex] showed six coordinated octahedral geometry, while [La(III), and Ce(III) complexes]were structured with coordination number seven.  Schiff's bases a

English is spoken by its native speakers in two different forms. Reduced form which marks the colloquial and rapid speech so that it is easily produced and a citation or unreduced form which is a characteristic of careful, emphasized and slow speech.

      This paper investigates Iraqi EFL university students’ production of the two forms mentioned above. The sample chosen includes twenty fourth year students, of which ten are males and the other ten are females from the Department of English of the College of Languages of the University of Duhok in Kurdistan Region of Iraq in the academic year 2020-2021.  The material tested is six connective words which represent the commonest ones in every-day co

Commercially pure titanium (cpTi) is widely used as dental implant material although it was found that titanium exhibited high modulus of elasticity and the lower corrosion tendency in oral environment. Niobium(Nb) was chosen for this study as an alternative to cpTi implant material due to its bioinert behavior and good elastic modulus and moderate cost in addition to corrosion resistance. This study was done to evaluate the effect of biomimetic coating on the surface properties of the commercially pure titanium and niobium implants by in vitro and in vivo experiments. The in vitro study was involved etching the samples of each material in HCl then soaking in 10M NaOH aqueous solution. These samples were then immersed in a 5 times concent

To determine the potential of gingival crevicular fluid (GCF) volume, E‐cadherin and total antioxidant capacity (TAC) levels to predict the outcomes of nonsurgical periodontal therapy (NSPT) for periodontitis patients.

A class of hyperrings known as divisible hyperrings will be studied in this paper. It will be presented as each element in this hyperring is a divisible element. Also shows the relationship between the Jacobsen Radical, and the set of invertible elements and gets some results, and linked these results with the divisible hyperring. After going through the concept of divisible hypermodule that presented 2017, later in 2022, the concept of the divisible hyperring will be related to the concept of division hyperring, where each division hyperring is divisible and the converse is achieved under conditions that will be explained in the theorem 3.14. At the end of this paper, it will be clear that the goal of this paper is to study the concept

In Indonesia, cattle feces (CF) and water hyacinth (WH) plants are abundant but have not been widely revealed. The use of microorganisms as decomposers in the fermentation process has not been widely applied, so researchers are interested in studying further. This study was to evaluate the effect of the combination of CF with WH on composting by applying white-rot fungal (WRF) (Ganoderma sp) microorganism as a decomposer. A number of six types of treatment compared to R₁(ratio of CF:WH)(25%:75%)+WRF; R₂(ratio of CF:WH)(50%:50%)+WRF; R₃(ratio of CF:WH)(75%:25%)+WRF; R₄(ratio of CF:WH)(25%:75%) without WRF; R₅(ratio of CF:WH)(50%:50%) without WRF; R₆(ratio of CF:WH)