A Short Note On Some Novel Applications of Semi Module Homomorphisms
Mohammad Abobala, Tishreen University, Department of Mathematics, Syria
Mikail Bal, Gaziantep University, Department of Mathematics, Turkey
Malath Aswad, Albaath University, Department of Mathematics, Syria
Co-: mohammadabobala777@gmail.com
Abstract
In this paper, we study the algebraic relationships between n- refined neutrosophic modules by using semi-module homomorphisms.
On the other hand, this work shows the relationship between neutrosophic geometrical AH-isometry and semi-module isomorphisms.
Keywords: AH-isometry, Semi homomorphism, semi isomorphism, n-refined neutrosophic module
1.Introduction
Neutrosophy is a new kind of philosophy founded by Smarandache [1] to study the uncertainty and the lack of information in many areas os science and life.
The indeterminacy element I and its refinements, were usefule in algebra where we find some generalizations of classical algebraic structures such as neutrosophic modules , n-refined neutrosophic matrices [2-5].
The concept of semi-modules homomorphism /isomorphism was presented in [6] as a tool to study the relations between modules which are defined over different rings.
In this work, we show a new application of semi homomorphisms in the study of n-refined neutrosophic modules. Also, we present a novel application in neutrosophic Euclidean geometry [9] based on semi- isomorphisms, where we show that algebraic isometries used in the study of neutrosophic Euclidean geometrical shapes can be considered as semi-module isomorphisms.
Main discussion
Firs of all, we show that the concept of semi-homomorphism (semi-isomorphism) is essential in the study of neutrosophic Euclidean geometry.
Neutrosophic Euclidean geometry wae built over the idea of AH-isometry, where the AH-isometry is a function that preserves distances between neutrosophic points in .
Definition 1.
(a). Let be the neutrosophic plane with two N-dimensions, then.
is called the two dimensional AH-isometry.
(b). the function ; is called the one dimensional AH-isometry.
Theorem 2
preserve operations and distances.
In the following, we prove that the two dimensional AH-isometry is a semi-module isomorphisim.
Definition 3:
Let ( M,+,.) be a module over the ring R, then (M(I),+,.) is called a weak neutrosophic module over the ring R, and it is called a strong neutrosophic module if it is a module over the neutrosophic ring R(I).
Elements of M(I) have the form , i.e M(I) can be written as .
Definition 4:
(a) Let (M,+,.) be a module over the ring R, we say that is a weak n-refined neutrosophic module over the ring R. Elements of are called n-refined neutrosophic vectors, elements of R are called scalars.
(b) If we take scalars from the n-refined neutrosophic ring , we say that is a strong n-refined neutrosophic module over the n-refined neutrosophic ring . Elements of are called n-refined neutrosophic scalars.
Definition 5:
Let M be a module over a ring R, N be a module over a ring T, be a well defined map, we say that is an -semi module homomorphism if and only if the following conditions are true:
(a) for all .
(b) There is a ring homomorphism such for all M.
Example 6:
Let M be a module over a ring R, M(I) be the corresponding strong neutrosophic module over R(I), be the corresponding strong refined neutrosophic module over . Then
is a semi homomorphism.
Theorem 7.
Let be the two dimensional AH-isometry, and be the one dimensional AH-isometry, then, is a semi-module isomorphism.
Proof.
we have is a module over the ring , and is a module over the ring .
According to [14], is a ring isomorphism.
On the other hand, preserves addition between and , and is a bijection.
Now, we prove that has the following property.
.
Thus is a semi isomorphism.
Result 8.
According to the previous theorem, the neutrosophic Euclidean geometry can be undersood as a semi isomorphic image of a neutrosophic module.
Now, we study semi homomorphisms between n-refined neutrosophic modules.
Theorem 9
Let be a ring, be the corresponding n-refined neutrosophic ring.
Let be a module over , be the corresponding n-refined neutrosophic module over , then.
(a). is a semi homomorphic image of .
(b). is a semi homomorphic image of .
Proof.
(a).
is a ring homomorphism.
We define .
It is clear that is well defined and preserves addition.
Now, Let's compute:
The coefficient of is .
The coefficient of is .
This implies that
Now, we compute.
Where
Which is exactly equal to the coefficient in .
This means that , hence is a semi module homomorphism.
(b). since is a semi homomorphic image of , we get the following sequence:
Thus is a semi homomorphic image of .
Example 10
Let . is a module over .
Let be the corresponding 3-refined neutrosophic module over
.
(a).
Is a ring homomorphism.
(b).
is a semi homomorphism, that is because:
preserves addition clearly.
On the other hand.
(c).
.
The symbol means semi isomorphic property.
Conclusion
In this paper, we have presented some novel applications of semi module homomorphisms/isomorphisms, where we proved that every module is a semi homomorphic image of it corresponding n-refined neutrosophic module.
Also, we have shown that the AH-isometry used in the theory of neutrosophic Euclidean geometry is a semi-module isomorphism.
References
[1] Smarandache, F., " A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability", American Research Press. Rehoboth, 2003.
[2] Abobala, M., Hatip, A., and Bal, M., " A Review On Recent Advantages In Algebraic Theory Of Neutrosophic Matrices", IJNS, Vol.17, 2021.
[3] Agboola, A.A.A., "On Refined Neutrosophic Algebraic Structures", NSS,Vol.10, pp. 99-101. 2015.
[4] Abobala, M., " A Study Of Nil Ideals and Kothe's Conjecture In Neutrosophic Rings", International Journal of Mathematics and Mathematical Sciences", Hindawi, 2021
[6] Abobala, M., " Semi Homomorphisms and Algebraic Relations Between Strong Refined Neutrosophic Modules and Strong Neutrosophic Modules", Neutrosophic Sets and Systems, Vol. 39, 2021.
[7] Abobala, M., "On Some Algebraic Properties of n-Refined Neutrosophic Elements and n-Refined Neutrosophic Linear Equations", Mathematical Problems in Engineering, Hindawi, 2021
[8] Abobala, M., "Neutrosophic Real Inner Product Spaces", NSS, Vol. 43, 2021.
[9] Abobala, M., and Hatip, A., "An Algebraic Approch To Neutrosophic Euclidean Geometry", NSS, Vol. 43, 2021.
[10] Abobala, M., "On The Characterization of Maximal and Minimal Ideals In Several Neutrosophic Rings", NSS , Vol. 45, 2021.