A Study Of Some Neutrosophic Clean Rings
Mohammad Abobala, Tishreen University, Department of Mathematics, Syria
Mikail Bal, Gaziantep University, Department of Mathematics, Turkey
Ahmed Hatip, Gaziantep University, Department of Mathematics, Turkey
Correspondance: mohammadabobala777@gmail.com
Abstract
The objective of this paper is to introduce a necessary and sufficient condition for a neutrosophic ring to be clean. This work proves the equivalence between case of classical clean ring and the corresponding neutrosophic ring ,refined neutrosophic ring , and n-refined neutrosophic ring .
Keywords: refined neutrosophic ring, n-refined neutrosophic ring ,classical clean ring, neutrosophic clean ring.
1.Introduction
Neutrosophy is a new kind of generalized logic founded by Smarandache to handle the indeterminacy and uncertainity in science and life.
The concept of neutrosophic rings was defined by Kandasamy and Samarandache in [1], as an extention of any algebraic ring .
These rings played an important role in algebraic studies such as as number theory [2,3],functional analysis [4], and algebraic geometry [5].
Recently, neutrosophic rings have been expanded to refined neutrosophic rings [6], and n-refined neutrosophic rings [7].
In [11-13], we find many theorems about the structure of idempotents and special sets of these rings.
In [9], the concept of neutrosophic clean ring was studied with many elementary interesting properties.
This motivates us to extend the previous works about clean rings to refined and n-refined neutrosophic rings, where we show the equivalence between clean conditions in any ring and its corresponding neutrosophic ring , refined neutrosophic ring , and n-refined neutrosophic ring . All rings are considered with unity 1.
Definition 1.[1]
Let R be a ring, I be the indeterminacy with property , then the neutrosophic ring is defined as follows:
.
Definition 2. [6]
(a) The element I can be split into two indeterminacies with conditions:
(b) If X is a set then X( is called the refined neutrosophic set generated by X , .
(c) Let (R,+, ) be a ring, (R( is called a refined neutrosophic ring generated by R , .
Definition 3.
Let (R,+, ) be a ring and be n indeterminacies. We define (I)={ } to be n-refined neutrosophic ring.
Addition and multiplication on (I) are defined as:
.
Where × is the multiplication defined on the ring R.
Definition 4. [10]
Let be any ring, be an arbitrary element, then.
a). is called a unit if there exists , such that .
b). is called an idempotent if and only if .
Theorem 5.[10]
Let a neutrosophic ring, then.
a). is idempotent if and only if are idempotents in .
b). is unit if and only if units in .
Theorem 6.[10]
Let be a refined neutrosophic ring, then.
a). is an idempotent if and only if are idempotents in .
b). is a unit if and only if are units in .
Definition 7. [9]
Let be a ring, then it is called clean if and only if for each , we have , where is a unit and is an idempotent.
Example 8.
is clean, that is because.
( .
( .
( .
3. Main discussion.
Theorem 3.1:
Let be any ring, be its corresponding neutrosophic ring.
is clean if and only if is clean.
Proof.
If is clean, then is clean. That is because .
Conversely, suppose that is clean, we must prove that is clean.
, then .
By the assumption, we can write.
; where are units and are idempotents.
Now, we have .
is a unit, that is because , are units in . Also, is an idempotent, that is because , are idempotents in .
[See theorem 5].
Thus is a sum of an idempotent with a unit, hence is a clean.
Example 3.2:
We have shown that is a clean ring.
According to the previous theorem is clean.
Remark that
. Where are units and
are idempotents.
Theorem 3.3:
Let be any ring, be its corresponding refined neutrosophic ring.
is clean if and only if is clean.
Proof.
is a homomorophic image of , hence if is clean, then is clean.
Conversely, assume that is clean, we must prove that is clean.
, we have.
, hence ; where are units and are idempotents.
Also, .
Bu using theorem 6, we get.
is a unit, that is because , are units in . Also, is an idempotent by a similar discussion.
This implies that is clean.
Example 3.4:
is clean ring, where
is clean too.
Remark that: .
The rest can be computed by the same way.
Theorem 3.5:
Let be any ring, be its corresponding n-refined neutrosophic ring. Then is clean if and only if is clean.
Proof.
If isclean, then is clean as a direct result of the homomorphism between and .
For the converse, suppose that is clean. We must prove that is clean.
=
According to theorem , is a unit, that is because , are units in .
By the same method, we get that is an idempotent, hence is clean.
4. Conclusion
In this article, we have determined necessary and sufficient condition for a neutrosophic ring to be clean. Also, we have found the equivalence between classical clean ring and the corresponding neutrosophic ring , refined neutrosophic ring , and n-refined neutrosophic ring .
Funding: “This research received no external funding”
Conflicts of Interest: “The authors declare no conflict of interest.”
References
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