On The 4-Cyclic Refined Neutrosophic Solutions of The Diophantine Equation   and m-Cyclic Refined Neutrosophic Modulo Integers

Lee Xu¹, Maretta Sarkis², Ammar Rawashdeh³, Ahmad Khaldi⁴

¹University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China

²Abu Dhabi University, Abu Dhabi, UAE

³Mutah University, Faculty of Science, Jordan

⁴Mutah University, Faculty of Science, Jordan,

Emails:  Leexu1244@yahoo.com; Sarkismaretta1990@gmail.com; ammarrawashde8932@gmail.com; khaldiahmad1221@gmail.com

Abstract

The ring of n-cyclic refined neutrosophic integers is a logical extension of the integer ring Z based on a special multiplication operation defined between the indeterminacy algebraic elements. In this paper, we provide a full description of the 4-cyclic refined neutrosophic integer roots of unity, where we prove that for odd values of n we get exactly two different solutions. For even values of n, we get exactly 15 different solutions. On the other hand, we characterize the m-cyclic refined neutrosophic modulo integers rings and present many of their algebraic properties based on neutrosophic homomorphisms and substructures.

Keywords: 4-cyclic refined neutrosophic integer; Diophantine equation; 4-cyclic refined neutrosophic solution; roots of unity; m-cyclic refined neutrosophic modulo integers ring.