A Contribution To Kothe's Conjecture In Refined Neutrosophic Rings

 

Arwa A. Hajjari,  Rozina Ali,

 

Cairo University, Egypt

 

Correspondance: rozyyy123n@gmail.com

 

                                                                              

 

Abstract 

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 .

Keywords: refined neutrosophic ring, Kothe Conjecture, Nilpotent Ideal.

1.Introduction

Neutrosophic algebra began with the work of Smarandache and Kandasamy in [6], where they presented concepts such as neutrosophic groups, neutrosophic rings and loops.

This motivates researchers to study a lot of extensions of classical algebraic structures by using neutrosophic ideas. See [2,3,4,17,46].

In [28,30], we find an interesting generalization of neutrosophic rings,  where the concept of refined neutrosophic rings was invented. Many interesting properties of these rings were discussed such as congruencies [10], partial ordering [44], AH-homomorphisms [15], and idempotency [45].

Refined neutrosophic rings were applicable in other algebraic structures, where refined neutrosophic modules [19], refined matrices [31,35], can be built over these rings in a similar way to classical cases.

In [9], Abobala has proved that every neutrosophic ring is a homomorphic image of the corresponding refined neutrosophic ring. 

Recently, the structure of maximal and minimal full ideals of refined neutrosophic rings was presented [5]. This structure implies an inclusion between the classical parts of this ideal.

In this article, we will show that inclusion's condition is not true in the case of rings with no unity. In particular, we determine the condition of any full ideal to be nil in any refined neutrosophic ring . Also, an equivalence between Kothe's conjecture in classical ring R and the corresponding refined neutrosophic ring  will be presented.

The motivation of our work is the open question listed in [42].

 

2. Preliminaries.

For the concepts of neutrosophic rings, refined neutrosophic rings, and ideals. See [30,32,42,45].

Kothe Conjecture:

If R is a ring, then the sum of any two left nil ideals is nil again.

Theorem [5]:

Consider the following:

  be a refined neutrosophic ring,  be a subset of  

. M is an ideal of  if and only if

(a)  are ideals on R

 .

Open problem [42]:

If Kothe's conjecture is true in the ring R, then is it true in the corresponding refined neutrosophic ring R( 

3. Main discussion.

In [5], authors have proved that   is a full ideal of  if and only if , under the condition that R has unity. First of all, we give an example to show that this statement is not true if R has not a unity.

Example 3.1:

Consider the refined neutrosophic ring of even integers, let  be a subset of it. First of all, we show that P is a full ideal of . It is easy to see that (P,+) is a subgroup. Let  be any element of P,  be any element of , we have:

 . Thus P is a full ideal and the inclusion's condition is not available, that is because 2Z is not contained in 4Z.

The following theorem describes the structure of nil ideals in .

Theorem 3.2:

Let  be any neutrosophic ring, we have:

(a)  is nilpotent in  if and only if  are nilpotent elements in R.

(b) If  is an ideal of , then P is nilpotent if and only if  are nilpotent.

(c) If  is a right/left ideal of , then P is nil if and only if  are nil.

Proof:

(a) In the beginning, we show that , where n is any positive integer.

For n=1 it is clear. We suppose that it is true for , we shall prove it for k+1.

  .

Hence  

Thus it is true by induction.

Now, we suppose that   is nilpotent in , hence there is a positive integer n such that

 . By the previous statement, we get . Thus  are nilpotent elements in R. The converse is easy.

(b) Let  be a nilpotent ideal of , there exists a positive integer n such that  .

For any  we have  , hence , and Q is nilpotent.

On the other hand for any  we have  , hence , and M is nilpotent . By the same, we get that N is nilpotent.

Now, for every , we have , by the assumption of the nilpotency of P, we get , hence , which implies that  are nilpotent.

The converse is easy and clear. 

(c) Let  be a nil ideal of , and  be an arbitrary element of P, then there exists a positive integer n such that  , thus , so that  are nil.

Also,  are nil ideals, that is because .

For the converse, we assume that  are nil ideals in the classical ring R, we must prove that  is a nil ideal of .

Let  be an arbitrary element of P, we have . There exists three positive integers  such that . Now, we compute

 . This means that P is a nil ideal of the refined neutrosophic ring .

The following theorem shows the equivalence between Kothe's conjecture in the classical ring R and the corresponding refined neutrosophic ring .

Theorem 3.4:

Kothe's conjecture is true in the refined neutrosophic ring  if and only if it is true in the corresponding classical ring R.

Proof:

If Kothe's conjecture is true in  , then it is true in R, that is because R is a homomorphic image of . See [9].

Now, suppose that Kothe's conjecture is true in . If  are two nil ideals of , then by theorem , we get  are nil in R, hence  is a nil ideal in , that is because

  are nil in R (Since the Kothe's conjecture is true in the ring R by the assumption). This implies that Kothe's conjecture is true in the refined neutrosophic ring .

4. Conclusion

In this article, we have proved the equivalence between Kothe's conjecture in classical rings and refined neutrosophic rings. Also, we have determined the necessary and sufficient condition for any ideal in  to be a nil ideal. On the other hand, we give a counter example to show that inclusion's condition between the parts any ideal in  is not available if R has not a unity.

Funding: “This research received no external funding” 

Conflicts of Interest: “The authors declare no conflict of interest.” 

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