Neutrosophic Ring Research Articles

Neutrosophic Topological Vector Spaces and its Properties

The algebraic structures Group, Ring, Field and Vector spaces are important innovations in Mathematics. Most of the theoretical concepts of Mathematics are based on the theorems related to these algebraic structures. Initially many mathematicians developed theorems related to all these algebraic structures. In 20th century most of the researchers introduced the theorems on the algebraic structures with Fuzzy and Intuitionistic fuzzy sets. Recently in 21st century the researchers concentrated on Neutrosophic sets and introduced the algebraic structures like Neutrosophic Group, Neutrosophic Ring, Neutrosophic Field, Neutrosophic Vector spaces and Neutrosophic Linear Transformation. In the current scenario of relating the spaces with the structures, we have introduced the concepts of Neutrosophic topological vector spaces. In this article, the study of Neutrosophic Topological vector spaces has been initiated. Some basic definitions and properties of classical vector spaces are generalized in Neutrosophic environment over a Neutrosophic field with continuous functions. Neutrosophic linear transformations and their properties are also included in Neutrosophic Topological Vector spaces. This article is an extension work of fuzzy and intuitionistic fuzzy vector spaces which were introduced in fuzzy and intuitionistic fuzzy environments. Even though it is an extension work, Neutrosophic Topological Vector space will play an important role in Neural Networks, Image Processing, Machine Learning and Artificial Intelligence Algorithms.

On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity

The 3-cyclic refined neutrosophic roots of unity are exactly the solutions of the Diophantine equation in the 3-cyclic refined neutrosophic ring of integers . This paper is dedicated to finding all 3-cyclic refined neutrosophic Diophantine roots of unity, where it proves that there exist only three solutions for the case of odd order (n), and twelve different solutions for the case of even order (n). On the other hand, the group generated from all solutions will be classified as a finite abelian group with direct products of finite cyclic groups.

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.

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. 

A Review Study on Some Properties of The Structure of Neutrosophic Ring

A Review Study on Some Properties of The Structure of Neutrosophic Ring

On Nil-clean Neutrosophic Rings

A ring is said to be nil-clean if every element of the ring can be written as a sum of an idempotent element and a nilpotent element of the ring. In this paper, we generalize this argument to neutrosophic structure. We introduce the structure of nil-clean neutrosophic ring and some of its elementary properties are presented. Also, we have found the equivalence between classical nil-clean ring R and the corresponding neutrosophic ring R(I), refined neutrosophic ring R(I1, I2), and n-refined neutrosophic ring Rn(I).

On the Solutions of Fermat's Diophantine Equation in 2-cyclic Refined Neutrosophic Ring of Integers

The Diophantine equation X^n+Y^n=Z^n is called the Fermat's Diophantine equation. Its solutions are called general Fermat's triples.The aim of this paper is to study the solutions of Fermat's Diophantine equation in the 2-cyclic refined neutrosophic ring of integers, where we determine all possible solutions of this Diophantine equation, as well as, the special case of Pythagoras triples.

A Study Of Some Neutrosophic Clean Rings

The objective of this paper is to introduce a necessary and sufficient condition for a neutrosophic ring to be clean. This work proves the equivalence between case of classical clean ring and the corresponding neutrosophic ring ,refined neutrosophic ring , and n-refined neutrosophic ring .

On the Conditions of Imperfect Neutrosophic Duplets and Imperfect Neutrosophic Triplets

In any neutrosophic ring R(I), an imperfect neutrosophic duplet consists of two elements x,y with a condition xy=yx=x and an imperfect neutrosophic triplet consists of three elements x,y,z with condition xy=yx=x,yz=zy=z,and xz=zx=y. The objective of this paper is to determine the necessary and sufficient conditions for neutrosophic duplets and triplets in any neutrosophic ring R(I), and to determine all triplets in Z(I).

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.

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).

A Study of Nil Ideals and Kothe’s Conjecture in Neutrosophic Rings

The aim of this study is to determine the necessary and sufficient condition for any AH subset to be a full ideal in a neutrosophic ring R(I) and to be a nil ideal too. Also, this work shows the equivalence between Kothe’s conjecture in classical rings and corresponding neutrosophic rings, i.e., it proves that Kothe’s conjecture is true in the neutrosophic ring R(I) if and only if it is true in the corresponding classical ring R.

On Some Algebraic Properties of n-Refined Neutrosophic Elements and n-Refined Neutrosophic Linear Equations

This paper studies the problem of determining invertible elements (units) in any n-refined neutrosophic ring. It presents the necessary and sufficient condition for any n-refined neutrosophic element to be invertible, idempotent, and nilpotent. Also, this work introduces some of the elementary algebraic properties of n-refined neutrosophic matrices with a direct application in solving n-refined neutrosophic algebraic equations.

A Contribution To Kothe's Conjecture In Refined Neutrosophic Rings

The objective of this paper is to answer an open question asked in [42], about the equivalence between Kothe's conjecture in a ring R and its corresponding refined neutrosophic ring . Where it proves that Kothe's conjecture is true in R if and only if it is true in .

On Imperfect Duplets In Some Refined Neutrosophic Rings

This paper solves the imperfect duplets problem in refined neutrosophic rings, where it presents the necessary and sufficient conditions for a pair to be an imperfect duplet in any refined neutrosophic ring. Also, this work introduces a full description of the structure of imperfect duplets in numerical refined neutrosophic rings such as refined neutrosophic ring of integers , refined neutrosophic ring of rationales , and refined neutrosophic ring or real numbers.

A Short Note On The Solutions Of Fermat's Diohantine Equation In Some Neutrosophic Rings

This paper is dedicated to study the concept of Fermat's triples in rings. Also, it determines the possible Fermat's triples in the neutrosophic ring of integers Z(I). Also, it discussed these triples in several finite commutative rings such as Zn.

Neutrosophic Linear Diophantine Equations With Two Variables

This paper is devoted to study for the first time the neutrosophic linear Diophantineequations with two variables in the neutrosophic ring of integers , and refined neutrosophicring of integers . This work introduces an algorithm to solve the linear Diophantineequation in , .

Classical Homomorphisms Between Refined Neutrosophic Rings and Neutrosophic Rings

The aim of this paper is to study homomorphisms between refined neutrosophic rings and neutrosophic rings. We prove that every neutrosophic ring R(I) is a homomorphic image of the refined neutrosophic ring R(I_1,I_2). Furthermore, we prove the following interesting result: