Characterization of Associative PU-algebras by the Notion of Derivations

Mehmood Khan; Dawood Khan; Khalida Mir Aalm

doi:doi:10.11648/j.ajmcm.20210601.13

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Characterization of Associative PU-algebras by the Notion of Derivations

Mehmood Khan, Dawood Khan, Khalida Mir Aalm

Published in American Journal of Mathematical and Computer Modelling (Volume 6, Issue 1)

Received: 18 January 2021 Accepted: 2 February 2021 Published: 4 March 2021

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Abstract

In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z- algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.

Published in	American Journal of Mathematical and Computer Modelling (Volume 6, Issue 1)
DOI	10.11648/j.ajmcm.20210601.13
Page(s)	14-18
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

PU-Algebras, (Left, Right)-derivations of PU-algebras, (Right, Left)-derivations of PU-algebras, Regular Derivations of PU-algebras.

References

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[2]	Iséki, K. (1978). An introduction to the theory of BCK- algebras. Math. Japonica, 23, 1-26.
[3]	Imai, Y., & Iseki, K. (1966). On axioms of Proportional calculi xiv proc. Japan Acad, 42, 19-22.
[4]	Neggers, J., Ahn, S. S., & Kim, H. S. (2001). On Q- algebras. International Journal of Mathematics and Mathematical Sciences, 27 (12), 749-757.
[5]	Megalai, K., & Tamilarasi, A. (2010). Classification of TM- algebra. IJCA Special Issue on “Computer Aided Soft Computing Techniques for Imaging and Biomedical Applications” CASCT.
[6]	Mostafa, Samy M., MA Abdel Naby, and A. I. Elkabany. "New view of ideals on PU-algebra." International Journal of Computer Applications 111.4, 18 pages, (2015). View at: Publisher Site \| Google Scholar.
[7]	Hvala, B. (1998). Generalized derivations in rings. Communications in Algebra, 26 (4), 1147-1166.
[8]	Bell, H. E., & Kappe, L. C. (1989). Rings in which derivations satisfy certain algebraic conditions. Acta Mathematica Hungarica, 53 (3-4), 339-346.
[9]	Bell, H. E., & Mason, G. (1987). On derivations in near- rings. In North-Holland Mathematics Studies (Vol. 137, pp. 31-35). North-Holland.
[10]	Brešar, M., & Vukman, J. (1990). On left derivations and related mappings. Proceedings of the American Mathematical Society, 110 (1), 7-16.
[11]	Bresar, M. (1991). On the distance of the composition of two derivations to the generalized derivations. Glasgow Mathematical Journal, 33 (1), 89-93.
[12]	Muhiuddin, G., & Al-Roqi, A. M. (2012). On-Derivations in BCI-Algebras. Discrete Dynamics in Nature and Society, 2012.
[13]	Ilbira, S., Firat, A., & Jun, Y. B. (2011). On symmetric bi- derivations of BCI-algebras. Applied Mathematical Sciences, 5 (57-60), 2957-2966.
[14]	Al-Kadi, D. (2017). On (f, g)-Derivations of G-Algebras.
[15]	Zhao, W. (2018). Some open problems on locally finite or locally nilpotent derivations and ε-derivations. Communications in Contemporary Mathematics, 20 (04), 1750056.
[16]	Alekseev, A. V., & Arutyunov, A. A. (2020). Derivations in semigroup algebras. Eurasian Mathematical Journal, 11 (2), 9-18.
[17]	Muhiuddin, G., & Al-Roqi, A. M. (2012). On (α, β)- derivations in BCI-algebras. Discrete Dynamics in Nature and Society, 2012.
[18]	Abujabal, H. A., & Al-Shehri, N. O. (2017). On left derivations of BCI-algebras. Soochow Journal of Mathematics, 33 (3), 435.

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APA Style

Mehmood Khan, Dawood Khan, Khalida Mir Aalm. (2021). Characterization of Associative PU-algebras by the Notion of Derivations. American Journal of Mathematical and Computer Modelling, 6(1), 14-18. https://doi.org/10.11648/j.ajmcm.20210601.13

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ACS Style

Mehmood Khan; Dawood Khan; Khalida Mir Aalm. Characterization of Associative PU-algebras by the Notion of Derivations. Am. J. Math. Comput. Model. 2021, 6(1), 14-18. doi: 10.11648/j.ajmcm.20210601.13

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AMA Style

Mehmood Khan, Dawood Khan, Khalida Mir Aalm. Characterization of Associative PU-algebras by the Notion of Derivations. Am J Math Comput Model. 2021;6(1):14-18. doi: 10.11648/j.ajmcm.20210601.13

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@article{10.11648/j.ajmcm.20210601.13,
  author = {Mehmood Khan and Dawood Khan and Khalida Mir Aalm},
  title = {Characterization of Associative PU-algebras by the Notion of Derivations},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {6},
  number = {1},
  pages = {14-18},
  doi = {10.11648/j.ajmcm.20210601.13},
  url = {https://doi.org/10.11648/j.ajmcm.20210601.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210601.13},
  abstract = {In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z- algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.},
 year = {2021}
}

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TY  - JOUR
T1  - Characterization of Associative PU-algebras by the Notion of Derivations
AU  - Mehmood Khan
AU  - Dawood Khan
AU  - Khalida Mir Aalm
Y1  - 2021/03/04
PY  - 2021
N1  - https://doi.org/10.11648/j.ajmcm.20210601.13
DO  - 10.11648/j.ajmcm.20210601.13
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 14
EP  - 18
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20210601.13
AB  - In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z- algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.
VL  - 6
IS  - 1
ER  -

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Author Information

Mehmood Khan

Department of Mathematics, University of Balochistan, Quetta, Pakistan
Dawood Khan

Department of Mathematics, University of Balochistan, Quetta, Pakistan
Khalida Mir Aalm

Department of Mathematics, University of Balochistan, Quetta, Pakistan

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Plain Text BibTeX RIS

APA Style

Mehmood Khan, Dawood Khan, Khalida Mir Aalm. (2021). Characterization of Associative PU-algebras by the Notion of Derivations. American Journal of Mathematical and Computer Modelling, 6(1), 14-18. https://doi.org/10.11648/j.ajmcm.20210601.13

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ACS Style

Mehmood Khan; Dawood Khan; Khalida Mir Aalm. Characterization of Associative PU-algebras by the Notion of Derivations. Am. J. Math. Comput. Model. 2021, 6(1), 14-18. doi: 10.11648/j.ajmcm.20210601.13

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AMA Style

Mehmood Khan, Dawood Khan, Khalida Mir Aalm. Characterization of Associative PU-algebras by the Notion of Derivations. Am J Math Comput Model. 2021;6(1):14-18. doi: 10.11648/j.ajmcm.20210601.13

Copy | Download

@article{10.11648/j.ajmcm.20210601.13,
  author = {Mehmood Khan and Dawood Khan and Khalida Mir Aalm},
  title = {Characterization of Associative PU-algebras by the Notion of Derivations},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {6},
  number = {1},
  pages = {14-18},
  doi = {10.11648/j.ajmcm.20210601.13},
  url = {https://doi.org/10.11648/j.ajmcm.20210601.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210601.13},
  abstract = {In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z- algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Characterization of Associative PU-algebras by the Notion of Derivations
AU  - Mehmood Khan
AU  - Dawood Khan
AU  - Khalida Mir Aalm
Y1  - 2021/03/04
PY  - 2021
N1  - https://doi.org/10.11648/j.ajmcm.20210601.13
DO  - 10.11648/j.ajmcm.20210601.13
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 14
EP  - 18
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20210601.13
AB  - In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z- algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.
VL  - 6
IS  - 1
ER  -

Copy | Download