Novel Applications Of Neutrosophic AH-Isometries To The Group Of Units Problem In Neutrosophic Rings and Refined Neutrosophic Rings
Katy D. Ahmad, Islamic University Of Gaza, Palestine
Mikail Bal, Gaziantep University, Turkey
Arwa A. Hajjari, Cairo University, Egypt.
Rozina Ali, Cairo University, Egypt
Co-: Katyon765@gmail.com
Abstract: The Objective of this paper is to study the group of units problem in two different kinds of neutrosophic structures (neutrosophic rings and refined neutrosophic rings), where we use the concept of AH-isometry to classify neutrosophic rings\refined neutrosophic rings as direct products of classical rings with itself. Also, this classification will lead to the algebraic structure of the corresponding group of units.
Keywords: Group of units, AH-isometry, neutrosophic ring, refined neutrosophic ring
1.Introduction
After the arrival of neutrosophic ideas by Smarandache [1], many authors used this ideas to study algebra and algebraic structures, where we find many neutrosophical algebraic structures as generalizations of the classical ones, such as neutrosophic rings [4-5] and refined neutrosophic rings [7], and neutrosophic matrix theory [8-9].
In classical ring theory, the group of units problem is one of the most famous open problems [6]. This problem can be considered as a question about the classification of the group of units corresponded with a given ring. We can find many previous works and contributions about this subject such as [18].
In the literature, we find many essential studies in the field of neutrosophic functions or functions between neutrosophic structures such as neutrosophic inner products [2], AH-geometrical isometry [14].
In neutrosophic ring theory, the group of units problem [21] is still uncovered in general. Through this paper, we will study the group of units problem in refined neutrosophic rings\ neutrosophic rings by using the concept of AH-isometry (algebraic projections between rings and spaces). This combination between neutrosophic functions and neutrosophic structures will lead to a classification property of these rings, then we can solve the group of units problem in these rings easily.
All rings through this paper are taken with unity 1.
Group of Units Problem In Neutrosophic Rings
First of all, we recall some helpful definitions and concepts about neutrosophic rings.
Definition 1:
Let R be a ring, I be the indeterminacy with property , then the neutrosophic ring is defined as follows:
.
Definition 2:
Define the one-dimensional isometry between the ring R(I) and the Cartesian product R×R as follows:
.
It is an algebraic isomorphism between R(I) and
Example 3:
For The ring the corresponding neutrosophic ring is
Definition 4:
Let be a neutrosophic ring with unity 1, be an arbitrary element in R(I). It is called invertible or (unit) if there exists such that .
Example 5:
Consider the neutrosophic ring of integers Z(I)={ }. We can see that is a unit, that is because
The following theorem determines the criteria of invertibility.
Theorem 6:
Let R be any ring with unity 1, R(I) be its corresponding neutrosophic ring. An arbitrary element is invertible if and only if in R.
Proof:
Let be an invertible element in R(I). There is ; .
Thus this means that , hence
, this implies that elements in R.
Conversely, suppose that elements in R, then there is , where
with .
Remark 7:
In a neutrosophic field K(I). The invertibility condition becomes as follows:
An arbitrary element is invertible if and only if .
That is because all nonzero elements in the field K are invertible.
Example 8:
Let be the neutrosophic ring of integers modulo 4.
is invertible because are invertible in . The inverse of is
.
Theorem 9:
Let R be any finite ring with unity 1, R(I) be its corresponding neutrosophic ring. The order of the group of units is equal to .
Proof:
Let U(R(I)) be the group of units of the neutrosophic ring R(I). For any element in , we have are invertible in R, hence we can choose ) by ways. Also, can be chosen by ways, hence we get way to choose ). This implies that the number of elements in is
The previous theorem gives us an aspect about the algebraic structure of in general. For example if we consider the neutrosophic field (which is a ring by classical meaning). It will has a group of units of order . It is logical to think that . So it is natural to think about a classification property between R(I) and .
This goal can be achieved by regarding the properties of the one-dimensional AH-isometry. See [66].
We check it by the following theorem.
Theorem 10:
Let R be any ring with unity 1, R(I) be its corresponding neutrosophic ring. Then .
Proof:
Consider the one dimensional AH-isometry
, it is an isomorphism between the rings R(I), and .
This means that .
Now, we are able to determine the algebraic structure of .
Theorem 11:
Let R be any ring with unity 1, R(I) be its corresponding neutrosophic ring. Then .
Proof:
The proof holds directly from the fact that in the previous theorem, so that .
Example 12:
Let be the neutrosophic ring of integers modulo 4. It is known that the group of units of the ring is { }and isomorphic to .
The group of units of is U( )={ }. We can see that , i.e. the order f any element is 2.
Group Of Units Problem In Refined Neutrosophic Rings
Definition 13:
The element I can be split into two indeterminacies with conditions:
Definition 14:
Let (R,+, ) be a ring, (R( is called a refined neutrosophic ring generated by R , .
Remark 15:
For the operations (addition and multiplication) on the refined neutrosophic ring. See [15].
Definition 16:
Let be a refined neutrosophic ring with unity (1,0,0), be an arbitrary element in R( ). It is called invertible or (unit) if there is , such that .
Example 17:
is the refined neutrosophic ring of integers.
The following theorem determines the criteria of invertibility.
Theorem 18:
Let be a refined neutrosophic ring. An arbitrary element is invertible if and only if .
Proof:
Suppose that is invertible, then there is .
= , this means
, which implies that
.
Conversely, if , then there is where
. Thus t is invertible.
Remark 19:
In a refined neutrosophic field . The invertibility condition becomes as follows:
An arbitrary element An arbitrary element is invertible if and only if .
That is because all nonzero elements in the field K are invertible.
Example 20:
Let be the refined neutrosophic ring of integers modulo 4.
is invertible because , are invertible in . The inverse of is
=
.
Theorem 21:
Let R be any ring with unity 1, be its corresponding neutrosophic ring. Then .
Proof:
We can define the refined one-dimensional AH-isometry between and the ring as follows:
,
Let be two refined neutrosophic elements, then
.
.
is a correspondence one-to-one, that is because , and for every
,there exists such that .
Thus, is isomorphism.
The inverse isomorphism of is .
Now, we are able to determine the algebraic structure of .
Theorem 22:
Let R be any ring with unity 1, be its corresponding refined neutrosophic ring. Then .
Proof:
The proof holds directly from the previous theorem.
Example 23:
The following example will show an effective algorithm to compute the units of any refined neutrosophic ring .
Let be the refined neutrosophic ring of integers modulo 4.
By the last theorem, the algebraic structure of the group of units of is:
.
Hence it has exactly 8 units. To find all units, we can use the inverse isomorphism for the one dimensional refined AH-isometry, as follows:
The units in the ring are the triples }.
By taking the inverse image with the inverse isomorphism, we get all units in .
The units are }.
Conclusion
In this paper, we have studied the group of units problem in neutrosophic rings and refined neutrosophic rings. We have used the AH-algebraic isometry to classify the previous rings into Cartesian products of classical rings with itself and to classify the corresponding groups of units.
On the other hand, we have presented an effective algorithm to compute units in these rings.
As a future research direction, we aim to find an AH-isometry to classify n-refined neutrosophic rings and to solve the group of units problem in such rings.
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