in recent years cryptography has played a big role especially in computer science for information security block cipher and public

Let R be a commutative ring with identity, and let M be a unitary left R-module. M is called Z-regular if every cyclic submodule (equivalently every finitely generated) is projective and direct summand. And a module M is F-regular if every submodule of M is pure. In this paper we study a class of modules lies between Z-regular and F-regular module, we call these modules regular modules.

Inˑthis work, we introduce the algebraic structure of semigroup with KU-algebra is called KU-semigroup and then we investigate some basic properties of this structure. We define the KU-semigroup and several examples are presented. Also,we study some types of ideals in this concept such as S-ideal,k- ideal and P-ideal.The relations between these types of ideals are discussed and few results for product S-ideals of product KU-semigroups are given. Furthermore, few results of some ideals in KU-semigroup under homomorphism are discussed.

We present the concept of maps Γ- periodi2 on Γ -near-ring S. Our main goal is to research and explore the presence and mapping traits such as h Γ –hom anti-Γ –hom, Γ –α-derivations of Γ -periodi2 on Γ- near-rings.

Abstract

The aim of the current research is to prepare an integrated learning program based on mathematics standards for the next generation of the NYS and to investigate its impact on the development of the teaching performance of middle school mathematics teachers and the future thinking skills of their students. To achieve the objectives of the research, the researcher prepared a list of mathematics standards for the next generation, which were derived from a list of standards. He also prepared a list of the teaching competencies required for middle school mathematics teachers in light of the list of standards, as well as clarified the foundations of the training program and its objectives and the mathematical

The study aims to investigate the degree of student teachers at Sultan Qaboos University acquired skills in teaching Arabic via a virtual micro-teaching lab, as well as to reveal the difficulties they faced and their development proposals. To do this, the researchers developed a questionnaire divided into four dimensions: planning, implementation, evaluation, and

ethical values for the teaching profession, in addition to two open-ended questions to identify difficulties and suggestions. It was administered to (30) student teachers. The results revealed that the average degree of student-teacher acquisition of skills was high in its four dimensions. It ranged between (39.2) to (82.2), while the overall average was (56.2).