Neutrosophic Rings Research Articles

On The Classification of 3-Cyclic/4-Cyclic Refined Neutrosophic Real and Rational Von Shtawzen's Group

This paper aims to solve the group of units problem for 3-cyclic real and rational refined neutrosophic rings and 4-cyclic real and rational refined neutrosophic rings, where ring isomorphisms between the realrational 3-cyclic and 4-cyclic refined neutrosophic rings and the direct product of some field extensions of Q and R. These isomorphisms will help in classifying the group of units of each studied ring in terms of direct products of classical well-known abelian groups. Also, we use the classification isomorphisms to determine all ideals in these classes of algebraic rings.

Computing Idempotent Elements In 3-Cyclic and 4-Cyclic Refined Neutrosophic Rings of Integers

An element X in a ring R is called idempotent if it equals its square. In this paper, we study the idempotent elements in the 3-cyclic refined neutrosophic ring of integers and 4-cyclic refined neutrosophic ring of integers, where we compute all idempotents in those two rings by solving many different linear Diophantine systems which are generated directly from their the algebraic structure. On the other hand, we use the same Diophantine systems to compute all 2-potent 3-cyclic, and 4-cyclic refined neutrosophic integer elements.

Applications of neutrosophic rings whose elements are regular

In this paper, we develop some rings in algebra by studying their neutrosophic property. We have also studied several relationships between these rings using the concept of neutrosophic. The neutrosophic center of neutrosophic regular ring is also neutrosophic regular ring. Neutrosophic subring of neutrosophic regular ring is also neutrosophic regular ring.

On Some Different Types of Algebraic Linear Equations In Several Different Symbolic 2-Plithogenic Rings

Plithogenic algebraic rings are considered novel generalizations of neutrosophic rings. This paper is dedicated to studying and solving many different types of symbolic 2-plithogenic linear algebraic equations where dual, split-complex, and weak fuzzy linear equations of symbolic 2-plithogenic type will be solved and discussed. Also, we illustrate many numerical examples to clarify the validity of our work.

On the Non-Commutative Logical Rings As Novel Extensions of Neutrosophic Rings

This paper uses some logical algebraic elements to extend any ring into a non-commutative ring containing the original ring with many generalized substructures and special elements. On the other hand, we study the substructures of non-commutative logical rings such as AH-homomorphisms and AH-ideals with many examples that explain their algebraic validity. Also, we discuss the possibility of solving a linear Diophantine equation with two variables in the non-commutative logical ring of integers, where we present an easy algorithm to solve this kind of generalized Diophantine equation.

Some Special Refined Neutrosophic Ideals in Refined Neutrosophic Rings: A Proof-of-Concept Study


 In this research, we created notions of a refined neutrosophic prime (completely prime, semiprime, and completely semiprime) ideal in a refined neutrosophic ring. If R(I1, I2) is a refined neutrosophic ring, then each ideal of R(I1, I2) has the form J+KI1+LI2 are ideals of the classical ring R. The objective of this work is to find the necessary and sufficient condition on classical ideals J, L and K that makes J+KI1+LI2 a prime (completely prime, semiprime, and completely semiprime) ideal in R(I1, I2). We studied some of the elementary properties of these concepts and the most important properties that link them. 

Theory on Duplicity of Finite Neutrosophic Rings

Theory on Duplicity of Finite Neutrosophic Rings

On The Roots of Unity in Several Complex Neutrosophic Rings

On The Roots of Unity in Several Complex Neutrosophic Rings

On The Classification of The Group of Units of Rational and Real 2-Cyclic Refined Neutrosophic Rings

On The Classification of The Group of Units of Rational and Real 2-Cyclic Refined Neutrosophic Rings

Algorithms for Computing Pythagoras Triples and 4-Tiples in Some Neutrosophic Commutative Rings

This paper is dedicated to study the number theoretical Pythagoras triples\4-tiples problem in several kinds of neutrosophic algebraic systems, where it finds an algorithm to find Pythagoras triples\4-tiples in commutative neutrosophic rings and refined neutrosophic rings too. Besides, the necessary and sufficient condition for a triple\4-tiple to be Pythagoras triple\4-tiple (quadruples) is obtained and proven in term of theorems. In addition, many numerical examples will be illustrated.

Characterization of γ-Single Valued Neutrosophic Rings and Ideals

Characterization of γ-Single Valued Neutrosophic Rings and Ideals

A Review Study On Neutrosophic AH-Algebraic Structures

Neutrosophic Algebraic structures are rich fields for researchers to get many interesting generalizations of classical and fuzzy structures. This Study is dedicated to give the interested reader some of special neutrosophic algebraic substructures of neutrosophic algebraic structures, especially AH-substructures in neutrosophic rings, spaces, modules, and their generalizations.

Solutions of Some Kandasamy–Smarandache Problems about Neutrosophic Complex Numbers and Group of Units’ Problem

Neutrosophic complex numbers are a novel interesting generalization of classical complex numbers. This study aims to study the neutrosophic complex infinite rings, where we classify those rings as direct products by using classical complex field C. Also, this work introduces full solutions for 18 Kandasamy–Smarandache open problems concerning these structures, as well as the group of unit’s famous problem, where the group of units (invertible elements) of a neutrosophic complex ring will be classified as a direct product of two classical groups of units.

On Some Open Problems About n-Cyclic Refined Neutrosophic Rings And Number Theory

The objective of this paper is to present 40 open problems about the n-cyclic refined neutrosophic rings. These questions concern the n-cyclic refined neutrosophic rings, n-cyclic refined neutrosophic number theory, and n-cyclic refined neutrosophic analysis. They will represent the future of the study of neutrosophic n-cyclic refined rings and the related structures.

A Short Contribution To Von Shtawzen's Abelian Group In n-Cyclic Refined Neutrosophic Rings

This paper is dedicated to study the group of units (Von Shtawzen's group) of some numerical n-cyclic refined neutrosophic rings such as integer, rational, and real case. Where we write the elements of these abelian groups by using equations derived from the values of some circulant numerical determinants.

A Contribution to The Group of Units' Problem in Some 2-Cyclic Refined Neutrosophic Rings

This paper is dedicated to study the group of units problem in some 2-cyclic refined neutrosophic rings, where it presents a full classification of the group of units in 2-cyclic refined neutrosophic ring of integers , 2-cyclic refined neutrosophic ring of integers modulo 2 and 3 , and 2-cyclic refined neutrosophic ring of real numbers , as direct products of the cyclic groups. On the other hand, this work provides a necessary and sufficient condition for the invertibility of any square 2-cyclic refined neutrosophic real matrix. Also, we will illustrate some examples to clarify the validity of our work.

On the Classification of n-Refined Neutrosophic Rings and Its Applications in Matrix Computing Algorithms and Linear Systems

This work is dedicated to classifying the general n-refined neutrosophic ring by building a ring isomorphism and the direct product of the corresponding classical rings with itself. On the other hand, we use the classification property to solve the problem of n-refined neutrosophic computing of Eigen values and Eigen vectors of an n-refined neutrosophic matrix. Also, it will be applied to solve n-refined linear systems and models.

The Structure Of Imperfect Triplets In Several Refined Neutrosophic Rings

This paper solves the imperfect triplets problem in refined neutrosophic rings, where it presents the necessary and sufficient conditions for a triple (x,y,z) to be an imperfect triplet in any refined neutrosophic ring. Also, this work introduces a full description of the structure of imperfect triplets in numerical refined neutrosophic rings such as refined neutrosophic ring of integers Z(I1,I2) , refined neutrosophic ring of rationales Q(I1,I2), and refined neutrosophic ring or real numbers (R(I1,I2).

Conjectures for Invertible Diophantine Equations Of 3-Cyclic and 4-Cyclic Refined Integers

The objective of this paper is to suggest two new conjectures concerning the invertible elements in 3-cyclic, and 4-cyclic refined neutrosophic rings of integers, where the invertibility condition shows that the solution of some Diophantine equations may determine the classification of the group of units of these algebraic rings.

Examples on Some Novel Diophantine Equations Derived from the Group of Units Problem in n-Cyclic Refined Neutrosophic Rings of Integers

The objective of this paper is to present a new class of Diophantine equations derived from the group of units problem of n-cyclic refined neutrosophic rings of integers by using homomorphisms between these rings and a finite Cartesian product ring of Z with itself. Also, this work provides many examples about this class and its solvability as a new application of neutrosophic algebraic structures in number theory.