Neutrosophic Ring Research Articles

Refined Neutrosophic Rings I

The notion of neutrosophic ring R(I) generated by the ring R and the indeterminacy component I was introduced for the first time in the literature by Vasantha Kandasamy and Smarandache in.12 Since then, fur-ther studies have been carried out on neutrosophic ring, neutrosophic nearring and neutrosophic hyperring see.1, 3, 4, 6–8 Recently, Smarandache10 introduced the notion of refined neutrosophic logic and neutrosophic set with the splitting of the neutrosophic components T, I, F into the form T1, T2, . . . , Tp; I1, I2, . . . , Ir; F1, F2, . . . , Fs where Ti, Ii, Fi can be made to represent different logical notions and concepts. In,11 Smarandache introduced refined neutrosophic numbers in the form (a, b1I1, b2I2, . . . , bnIn) where a, b1, b2, . . . , bn ∈ R or C. The concept of refined neutrosophic algebraic structures was introduced by Agboola in5 and in particular, refined neutrosophic groups and their substructures were studied. The present paper is devoted to the study of refined neutrosophic rings and their substructures. It is shown that every refined neutrosophic ring is a ring. For the purposes of this paper, it will be assumed that I splits into two indeterminacies I1 [contradiction (true (T) and false (F))] and I2 [ignorance (true (T) or false (F))]. It then follows logically that:

Classical Homomorphisms Between n-Refined Neutrosophic Rings

This paper studies classical homomorphisms between n-refined neutrosophic ring and m-refined neutrosophic ring. It proves that every m-refined neutrosophic ring is a homomorphic image of n-refined neutrosophic ring , where . Also, it presents a discussion of kernels and some corresponding isomorphisms between those rings.

On Some Special Substructures of Neutrosophic Rings and Their Properties

In this paper we introduce the notions of AH-ideal and AHS-ideal as new kinds of neutrosophic substructures defined in a neutrosophic ring. We investigate the properties of these substructures and some related concepts as AH-weak principal ideal, AH-weak prime ideal and AH-weak maximal ideal.

Refined Neutrosophic Rings II

This paper is the continuation of the work started in the paper titled “Refined Neutrosophic Rings I”. In the present paper, we study refined neutrosophic ideals and refined neutrosophic homomorphisms along their elementary properties. It is shown that if R = Z(I1, I2) is a refined neutrosophic ring of integers and J = nZ(I1, I2) is a refined neutrosophic ideal of R, then R/J Zn(I1, I2).

N-Cyclic Refined Neutrosophic Algebraic Systems of Sub-Indeterminacies, An Application to Rings and Modules

Refining the indeterminate I into many levels of indeterminacy is a way to explore many neutrosophic algebraic structures.This paper introduces the concept of n-cyclic refined algebraic system of sub-indeterminacies as a new way to refine a neutrosophic indeterminate I. This idea will be used to introduce the notion of n-cyclic refined neutrosophic ring and to study its AH-substructures. Also, this work presents the concept of n-cyclic refined neutrosophic modules with many related structures.

N- Refined Neutrosophic Rings

The aim of this paper is to introduce the concept of n-refined neutrosophic ring as a generalization of refined neutrosophic ring. Also, wepresent concept of n-refined polynomial ring. We study some basic concepts related to these rings such as AH-subrings, AH-ideals, AH-factors, and AH-homomorphisms.

Neutrosophic Triplets in Neutrosophic Rings

The neutrosophic triplets in neutrosophic rings ⟨ Q ∪ I ⟩ and ⟨ R ∪ I ⟩ are investigated in this paper. However, non-trivial neutrosophic triplets are not found in ⟨ Z ∪ I ⟩ . In the neutrosophic ring of integers Z ∖ { 0 , 1 } , no element has inverse in Z. It is proved that these rings can contain only three types of neutrosophic triplets, these collections are distinct, and these collections form a torsion free abelian group as triplets under component wise product. However, these collections are not even closed under component wise addition.

Semi-Idempotents in Neutrosophic Rings

In complex rings or complex fields, the notion of imaginary element i with i 2 = − 1 or the complex number i is included, while, in the neutrosophic rings, the indeterminate element I where I 2 = I is included. The neutrosophic ring ⟨ R ∪ I ⟩ is also a ring generated by R and I under the operations of R. In this paper we obtain a characterization theorem for a semi-idempotent to be in ⟨ Z p ∪ I ⟩ , the neutrosophic ring of modulo integers, where p a prime. Here, we discuss only about neutrosophic semi-idempotents in these neutrosophic rings. Several interesting properties about them are also derived and some open problems are suggested.

On clean neutrosophic rings

A commutative ring is said to be clean if every element of the ring can be written as a sum of a unit and an idempotent. In this paper, we generalize this argument to structure of neutrosophic. We present the structure of clean neutrosophic ring. Some elementary properties of clean neutrosophic ring are also presented.

MODUL NEUTROSOFIK KUAT

Given any neutrosophic ring with unity and a commutatively additive neutrosophic group. Then we can formed a neutrosophic algebraic structure is called a strong neutrosphic module. From the concept of weak neutrosophic module we extend to the concept of strong neutrosophic module. In this paper, also elementary properties of strong neutrosophic module are given.

Soft Neutrosophic Ring and Soft Neutrosophic Field

In this paper, we extend the theory of neutrosophic rings and neutrosophic fields to soft sets and construct soft neutrosophic rings and soft neutrosophic fields. We also extend neutrosophic ideal theory to form soft neutrosophic ideal over a neutrosophic ring and soft neutrosophic ideal of a soft neutrosophic ring . We have given many examples to illustrate the theory of soft neutrosophic rings and soft neutrosophic fields and display many properties of of these. At the end of this paper, we gave soft neutrosophic ring homomorphism

Soft Nuetrosophic Ring and Soft Neutrosophic Field

In this paper, we extend the theory of neutrosophic rings and neutrosophic fields to soft sets and construct soft neutrosophic rings and soft neutrosophic fields. We also extend neutrosophic ideal theory to form soft neutrosophic ideal over a neutrosophic ring and soft neutrosophic ideal of a soft neutrosophic ring . We have given many examples to illustrate the theory of soft neutrosophic rings and soft neutrosophic fields and display many properties of of these. At the end of this paper, we gave soft neutrosophic ring homomorphism