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

Mohammad Abobala,R Ali Borzooei

doi:10.1155/2021/5573072

Abstract

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.

Highlights

Neutrosophy is a new kind of generalized logic proposed by Smarandache [1]
The relations between neutrosophic matrices and equations are defined in [24]. From this point of view, we are motivated to generalize the previous studies so that we study some of the algebraic properties of n-refined neutrosophic elements such as invertibility, nilpotency, and idempotency
(a) Let Rn(I) be an n-refined neutrosophic ring and P 􏽐ni 0 PiIi 􏼈a0 + a1I + · · · + anIn: ai ∈ Pi􏼉 where Pi is a subset of R; we define P to be an AH-subring if Pi is a subring of R for all i; AHS-subring is defined by the condition Pi Pj for all i, j

Summary

Introduction

Neutrosophy is a new kind of generalized logic proposed by Smarandache [1]. It becomes a useful tool in many areas of science such as number theory [2, 3], solving equations [4], and medical studies [5, 6]. Smarandache and Abobala proposed n-refined neutrosophic rings [16], modules [17, 18], and spaces [19,20,21,22]. The relations between neutrosophic matrices and equations are defined in [24]. From this point of view, we are motivated to generalize the previous studies so that we study some of the algebraic properties of n-refined neutrosophic elements such as invertibility, nilpotency, and idempotency. We study elementary properties of n-refined neutrosophic matrices and their application in solving the n-refined neutrosophic linear system of equations as a new generalization of previous efforts in [23,24,25]

Preliminaries

Main Discussion

Conclusion