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

Authors: Mohammad Abobala
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper studies the problem of determining invertible elements (units) in anyn-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.


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## Body

## 1. 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]. Also, we find many applications of neutrosophic structures in statistics [7], optimization [8], topology [9], and decision making [10, 11].On the other hand, neutrosophic algebra began in [12], where Smarandache and Kandasamy defined concepts such as neutrosophic groups and neutrosophic rings. These notions were handled widely by Agboola et al. in [13, 14], where homomorphisms and AH-substructures were studied [15].Recently, there is an increasing interest by the generalizations of neutrosophic algebraic structures. Smarandache and Abobala proposed n-refined neutrosophic rings [16], modules [17, 18], and spaces [19–22].Neutrosophic algebraic equations are useful in many scientific areas; there is a full description of their solutions in neutrosophic fields and refined neutrosophic fields [23]. In particular, 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. Also, 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 [23–25].


## 2. Preliminaries

Definition 1.
(see [16]).
Let (R, +, ×) be a ring and Ik,1≤k≤n be n indeterminacies. We define RnI=a0+a1I+⋯+anIn;ai∈R to be n-refined neutrosophic ring. If n = 2, we get a ring which is isomorphic to 2-refined neutrosophic ring RI1,I2.
Addition and multiplication onRnI are defined as follows:(1)∑i=0nxiIi+∑i=0nyiIi=∑i=0nxi+yiIi,∑i=0nxiIi×∑i=0nyiIi=∑i,j=0nxi×yjIiIj,where × is the multiplication defined on the ring R and xI0=x for all x∈RIjIi=IiIj=Imini,j,I0Ij=Ij.
It is easy to see thatRnI is a ring in the classical concept and contains a proper ring R.Definition 2.
(see [16]).
LetRnI be an n-refined neutrosophic ring, and it is said to be commutative if xy=yx for each x,y∈RnI; if there is 1∈RnI such 1⋅x=x⋅1=x, then it is called an n-refined neutrosophic ring with unity.Theorem 1 (see [16]).
LetRnI be an n-refined neutrosophic ring. Then,(a)
R is commutative if and only if RnI is commutative(b)
R has unity if and only if RnI has unity(c)
RnI=∑i=0nRIi=∑i=0nxiIi:xi∈RDefinition 3 (see [16]).
(a)
LetRnI be an n-refined neutrosophic ring and P=∑i=0nPiIi=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.(b)
P is an AH-ideal if Pi is a two-sided ideal of R for all i, and the AHS-ideal is defined by the condition Pi=Pj for all i,j.(c)
The AH-idealP is said to be null if Pi=RorPi=0 for all i.Definition 4.
(see [16]).
LetRnI be an n-refined neutrosophic ring and P=∑i=0nPiIi be an AH-ideal; we define AH-factor RI/P=∑i=0nR/PiIi=∑i=0nxi+PiIi;xi∈R.Theorem 2 (see [16]).
LetRnI be an n-refined neutrosophic ring and P=∑i=0nPiIi be an AH-ideal;
RnI/P is a ring with the following two binary operations:(2)∑i=0nxi+PiIi+∑i=0nyi+PiIi=∑i=0nxi+yi+PiIi,∑i=0nxi+PiIi×∑i=0nyi+PiIi=∑i=0nxi×yi+PiIi.


## 3. Main Discussion

In this section, we study the invertibility of any element in any n-refined neutrosophic ring, and we show the conditions of idempotency and nilpotency in these rings. All rings in this section are considered with unity 1.Definition 5.
LetX=A0+A1I1+⋯+AnIn be an n-refined neutrosophic element; we define its canonical sequence as follows:(3)M0=A0,Mj=A0+Aj+Aj+1+⋯+An,1≤j≤n.
For example,(4)M3=A0+A3+A4+⋯+An.Remark 1.
The multiplication operation between two n-refined neutrosophic elements can be represented by the following equation:(5)A0+A1I1+⋯+AnInB0+B1I1+⋯+BnIn=M0N0+MnNn−M0N0In+∑i=1n−1MiNi−Mi+1Ni+1Ii,where MiandNi are the canonical sequences of A0+A1I1+⋯+AnIn and B0+B1I1+⋯+BnIn, respectively.Proof.
Forn=0, the statement is true easily. Suppose that it is true for n=k, we must prove it for n=k+1. We compute the multiplication L=A0+A1I1+⋯+Ak+1Ik+1B0+B1I1+⋯+Bk+1Ik+1.(6)A0+A1I1+⋯+Ak+1Ik+1B0+B1I1+⋯+Bk+1Ik+1=A0+A1I1+⋯+AkIkB0+B1I1+⋯+BkIk+Ak+1Ik+1B0+B1I1+⋯+BkIk+A0+A1I1+⋯+AkIkBk+1Ik+1+Ak+1Ik+1Bk+1Ik+1=M0N0+MkNk−M0N0Ik+∑i=1kMiNi−Mi+1Ni+1Ii+I1Ak+1B1+A1Bk+1+I2Ak+1B2+A2Bk+1+⋯+IkAk+1Bk+AkBk+1+Ik+1A0Bk+1+Ak+1B0+Ak+1Bk+1.
Thus, the coefficient ofIk+1 is A0Bk+1+Ak+1B0+Ak+1Bk+1=Ak+1+A0Bk+1+B0−A0B0=Mk+1Nk+1−M0N0. Also, the coefficient of Ii, 1≤i≤k is MiNi−Mi+1Ni+1+Ak+1Bi+AiBk+1=A0+Ai+Ai+1+⋯+AkB0+Bi+Bi+1+⋯+Bk−A0+Ai+1+Ai+2+⋯+AkB0+Bi+1+Bi+2+⋯+Bk+Ak+1Bi+AiBk+1=A0+Ai+Ai+1+⋯+Ak+Ak+1B0+Bi+Bi+1+⋯+Bk+Bk+1−A0+Ai+1+Ai+2+⋯+Ak+Ak+1B0+Bi+1+Bi+2+⋯+Bk+Bk+1=MiNi−Mi+1Ni+1, where 1≤i≤k+1. Hence, our proof is completed by induction.Theorem 3.
.
LetX=A0+A1I1+⋯+AnIn be an n-refined neutrosophic element, then it is invertible if and only if Mj,0≤j≤n are invertible. The inverse of X is X−1=M0−1+Mn−1−M0−1In+∑j=1n−1Mj−1−Mj+1−1Ij=A0−1+A0+A1+⋯+An−1−A0+A2+⋯+An−1I1+A0+A2+⋯+An−1−A0+A3+⋯+An−1I2+A0+A3+⋯+An−1−A0+A4+⋯+An−1I3+⋯+A0+An−1−A0−1In.Proof.
X is invertible if and only if there exists Y=B0+B1I1+⋯+BnIn, where XY=YX=1. By using Remark 14, we can write the following:
M0N0+MnNn−M0N0In+∑i=1n−1MiNi−Mi+1Ni+1Ii=1. This implies thatM0N0=1 and MiNi−Mi+1Ni+1=0 for all i, where 0 is the zero element. Hence, we get MiNi=Mi+1Ni+1=M0N0=1. So, Mj,0≤j≤n are invertible.
On the other hand, we putX−1=M0−1+Mn−1−M0−1In+∑j=1n−1Mj−1−Mj+1−1Ij, and now we compute XX−1 as follows:(7)XX−1=M0M0−1+M1M1−1−M2M2−1I1+M2M2−1−M3M3−1I2+⋯+MnMn−1−M0M0−1In=1.Example 1.
.
ConsideringZI=a+bI1+cI2;a,b,c∈Z2 the 2-refined neutrosophic ring of integers, the set of invertible elements in Z2 is −1,1. Hence, the set of all invertible elements in the corresponding 2-refined neutrosophic ring is 1,−1,1−2I2,−1+2I2,1−2I1,−1+2I1,1+2I1−2I2,−1−2I1+2I2.Theorem 4.
.
LetX=A0+A1I1+⋯+AnIn be an n-refined neutrosophic element, and we have the following:(a)
X is nilpotent if and only if Mjfor allj are nilpotent(b)
X is idempotent if and only if Mjfor allj are idempotentProof.
(a)
First of all we will prove thatXr=M0r+InMnr−M0r+∑i=1n−1Mir−Mi+1rIi.We use the induction, forr=1 it is clear. Suppose that it is true for r=k, we prove it for k+1.(8)Xk+1=XkX=M0k+InMnk−M0k+∑i=1n−1Mik−Mi+1kIiA0+A1I1+⋯+AnIn=M0k+InMnk−M0k+∑i=1n−1Mik−Mi+1kIiM0+Mn−M0In+∑i=1n−1Mi−Mi+1=M0kM0+InMnkMn−M0kM0+∑i=1n−1MikMi−Mi+1kMi+1Ii=M0k+1+InMnk+1−M0k+1+∑i=1n−1Mik+1−Mi+1k+1Ii.X is nilpotent if there is a positive integer r such that Xr=0. This is equivalent to (9)M0r=Mnk=Mjk=0for allj,which implies the proof.(b)
The proof is similar to (a).


## 4.n-Refined Neutrosophic Linear Algebraic Equations

This section is dedicated to introduce an algorithm to solve n-refined neutrosophic linear equations over any n-refined neutrosophic field by turning them into classical systems of numbers.Also, we discuss some elementary properties of n-refined neutrosophic matrices.Definition 6.
.
LetFnI be any n-refined neutrosophic field. The n-refined linear neutrosophic equation with one variable over FnI is defined as follows:(10)AX+B=0,A,B,X∈FnI,where(11)A=a0+a1I1+⋯+anIn,B=b0+b1I1+⋯+bnIn,X=x0+x1I1+⋯+xnIn.Theorem 5.
.
LetFnI be any n-refined neutrosophic field and ∗AX+B=0 be any n-refined linear neutrosophic equation over FnI. Then, ∗ is solvable over FnI if and only if the following classical system is solvable over the classical field F:(1)
a0x0+b0=0(2)
a0+anx0+xn+b0+bn=0(3)
a0+an+an−1x0+xn+xn−1+b0+bn+bn−1=0..(n + 1 −) a0+a1+⋯+anx0+x1+⋯+xn+b0+b1+⋯+bn=0Proof.
We will show that Equation (18) is equivalent to the previous classical system of equations.
We compute Equation (18) by using the canonical form, and we get(12)M0N0+MnNn−M0N0In+∑i=1n−1MiNi−Mi+1Ni+1Ii=−b0−b1I1−⋯−bnIn,where MiandNi are the canonical forms of A and X, respectively.
From (12), we get the following classical system:(13)M0N0=−b0,MnNn−M0N0=−bn,MiNi−Mi+1Ni+1=−bi,forall1≤i≤n−1.
The equationM0N0=−b0 equivalents a0x0+b0=0. The equation MnNn−M0N0=−bn equivalents a0+anx0+xn+b0+bn=0.
Also, any equation with formMiNi−Mi+1Ni+1=−biforall1≤i≤n−1 equivalents a0+an+an−1+⋯+aix0+xn+xn−1+⋯+xi+b0+bn+bn−1+⋯+bi=0 by mathematical induction; thus, our proof is complete.
Now, we can apply the previous theorem to solve n-refined neutrosophic linear equations, and we illustrate an example.Example 2.
.
LetR be the real field and R3I be its corresponding 3-refined neutrosophic field. Consider the following 3-refined neutrosophic Equation (18) 1+I2+I3X+I1+2I2=0. To solve it, we turn it into the classical equivalent system.(1)
1⋅x0+0=0; its solution x0=0.(2)
1+1x0+x3+0+0=0; its solution is x0+x3=0;thusx3=0.(3)
1+1+1x0+x3+x2+0+0+2=0; its solution is 3x0+x3+x2=−2;thusx2=−2/3.(4)
1+1+1+0x0+x3+x2+x1+0+1+2+0=0; its solution is x0+x3+x2+x1=−1;thusx1=−1/3.
Hence, the solution of Equation (18) is(14)X=−23I2−13I1.Definition 7.
.
LetA=a11….a1m⋮⋱⋮an1…anm be an m×n matrix; if aij=x+yI1+zI2+⋯+tIn∈RnI, then it is called an n-refined neutrosophic matrix, where RnI is an n-refined neutrosophic ring.Remark 2.
.
IfA is an m×n matrix, then it can be represented as an element of the n-refined neutrosophic ring of matrices like the following: A=B+CI1+DI2+⋯+KIn where D,B,C,…,K are classical matrices with elements from the ring R and from size m×n.
For example,A=2+I1+3I2−I31−I1−I23+4I2+2I31+I1=2131+1−101I1+3−140I2+I3−1020 is a 3-refined neutrosophic matrix.Remark 3.
The identity with respect to multiplication is the normal unitary matrix.Definition 8.
.
LetA be a square m×mn-refined neutrosophic matrix, then it is called invertible if there exists an n-refined square m×m neutrosophic matrix B such that AB=Um×m where Um×m is the unitary classical matrix.Remark 4.
LetX=A0+A1I1+⋯+AnIn be a square m×m n-refined neutrosophic matrix, then it is invertible if and only if Mj,0≤j≤n are invertible. The inverse of X is(15)X−1=M0−1+Mn−1−M0−1In+∑j=1n−1Mj−1−Mj+1−1Ij=A0−1+A0+A1+⋯+An−1−A0+A2+⋯+An−1I1+A0+A2+⋯+An−1−A0+A3+⋯+An−1I2+A0+A3+⋯+An−1−A0+A4+⋯+An−1I3+⋯+A0+An−1−A0−1In.
The proof holds directly as a special case of Theorem3.Definition 9.
.
We defined the determinant of a squarem×mn-refined neutrosophic matrix as(16)detX=detA0+detA0+A1+⋯+An−detA0+A2+⋯+AnI1+detA0+A2+⋯+An−detA0+A3+⋯+AnI2+⋯+detA0+An−detA0In=detM0+detMn−detM0In+∑i=1n−1detMi−detMi+1Ii.
This definition is supported by the condition of invertibility.Theorem 6.
.
LetX=A0+A1I1+⋯+AnIn be a square m×mn-refined neutrosophic matrix, and we have the following:(a)
X is invertible if and only if detX≠0(b)
IfY=B0+B1I1+⋯+BnIn is a square m×mn-refined neutrosophic matrix, then detXY=detXdetY(c)
detX−1=detX−1Proof.
(a)
IfdetX≠0, this will be equivalent to detMj≠0for allj, i.e., Mj are invertible; thus, X is invertible according to Theorem 3.(b)
XY=M0N0+MnNn−M0N0In+∑i=1n−1MiNi−Mi+1Ni+1Ii. Hence,(17)detXY=detM0N0+IndetMnNn−detM0N0+∑i=1n−1detMiNi−detMi+1Ni+1Ii=detM0detN0+IndetMndetNn−detM0detN0+∑i=1n−1detMidetNi−detMi+1detNi+1Ii=detM0+detMn−detM0In+∑i=1n−1detMi−detMi+1IidetN0+detNn−detN0In+∑i=1n−1detNi−detNi+1Ii=detXdetY.(c)
It holds directly from (b).
Now, we can find an easy algorithm to solve a linear system of n-refined neutrosophic algebraic equations over any n-refined neutrosophic field by using the inverse matrix method.
We construct an example.Example 3.
.
Consider the following system of 2-refined neutrosophic linear equations:(18)2+I1+3I2X+1−I1−I2Y=−I1,(19)3+4I2X+1+I1Y=I2.
The corresponding refined neutrosophic matrix isA=2+I1+3I21−I1−I23+4I21+I1.
We have the following:(a)
A=2+I1+3I21−I1−I23+4I21+I1=2131+1−101I1+3−140I2 where B=2131, C=1−101, and D=3−140. B+D=5071,B+C+D=6−172.(b)
B−1=−113−2,B+D−1=1/50−7/51,B+C+D−1=2/191/19−7/196/19.(c)
A−1=B−1+I1B+C+D−1−B+D−1+I2B+D−1−B−1=−113−2+I1−9/951/1998/95−13/19+I26/5−1−22/53=−1−9/95I1+6/5I21+1/19I1−I23+98/95I1−22/5I2−2−13/19I1+3I2.It is easy to find thatA−1A=AA−1=1001.(d)
detB=−1,detB+D=5,detB+C+D=19,detA=−1+I119−5+I25−−1=−1+14I1+6I2.
SinceA is invertible, we get the solution of the previous system of the 2-refined linear system by computing the product:(20)A−1−I1I2=−1−995I1+65I21+119I1−I23+9895I1−225I2−2−1319I1+3I2,−I1I2=I11+995−65+119I1−3−9895+225−1319+I2−2+3,=−I1119−619I1+I2.
Thus,(21)X=−119I1,Y=−619I1+I2.


## 5. Conclusion

In this paper, we have determined the necessary and sufficient conditions for the invertibility, nilpotency, and idempotency of elements in a refined neutrosophic ring. In particular, we have studied some of algebraic properties of n-refined neutrosophic matrices such as determinants and inverses with an application solving the n-refined neutrosophic linear algebraic system of equations.As a future research direction, we aim to study n-refined neutrosophic Diophantine equations.



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