The Structure of Idempotents in Neutrosophic Rings and Neutrosophic Quadruple Rings

Yingcang Ma,Juanjuan Zhang,Florentin Smarandache,Xiaohong Zhang

doi:10.3390/sym11101254

Abstract

This paper aims to reveal the structure of idempotents in neutrosophic rings and neutrosophic quadruple rings. First, all idempotents in neutrosophic rings ⟨ R ∪ I ⟩ are given when R is C , R , Q , Z or Z n . Secondly, the neutrosophic quadruple ring ⟨ R ∪ T ∪ I ∪ F ⟩ is introduced and all idempotents in neutrosophic quadruple rings ⟨ C ∪ T ∪ I ∪ F ⟩ , ⟨ R ∪ T ∪ I ∪ F ⟩ , ⟨ Q ∪ T ∪ I ∪ F ⟩ , ⟨ Z ∪ T ∪ I ∪ F ⟩ and ⟨ Z n ∪ T ∪ I ∪ F ⟩ are also given. Furthermore, the algorithms for solving the idempotents in ⟨ Z n ∪ I ⟩ and ⟨ Z n ∪ T ∪ I ∪ F ⟩ for each nonnegative integer n are provided. Lastly, as a general result, if all idempotents in any ring R are known, then the structure of idempotents in neutrosophic ring ⟨ R ∪ I ⟩ and neutrosophic quadruple ring ⟨ R ∪ T ∪ I ∪ F ⟩ can be determined.

Highlights

The notions of neutrosophic set and neutrosophic logic were proposed by Smarandache [1].In neutrosophic logic, every proposition is considered by the truth degree T, the indeterminacy degree I, and the falsity degree F, where T, I and F are subsets of the nonstandard unit interval]0−, 1+ [= 0− ∪ [0, 1] ∪ 1+ .Using the idea of neutrosophic set, some related algebraic structures have been studied in recent years
We can obtain all idempotents in neutrosophic ring h R ∪ I i if all idempotents in any ring R are known
We study the idempotents in neutrosophic ring h R ∪ I i and neutrosophic quadruple ring h R ∪ T ∪ I ∪ F i

Summary

Introduction

The notions of neutrosophic set and neutrosophic logic were proposed by Smarandache [1]. As an application, if R = F, where F is any field, we can divide the elements of h R ∪ I i (or h R ∪ T ∪ I ∪ F i) by idempotents As another application, in paper [22], the authors explore the idempotents and semi-idempotents in neutrosophic ring hZn ∪ I i and some open problems and conjectures are given. The open problem and conjectures proposed in paper [22] about idempotents in neutrosophic ring hZn ∪ I i will be solved.

Basic Concepts

The Idempotents in Neutrosophic Rings

The Idempotents in Neutrosophic Quadruple Rings

Conclusions