On Commutativity of Rings Under Certain Polynomial Constraints

Mohammad Shadab Khan

doi:doi:10.11648/j.ijdsa.20180401.13

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On Commutativity of Rings Under Certain Polynomial Constraints

Mohammad Shadab Khan

Published in International Journal of Data Science and Analysis (Volume 4, Issue 1)

Received: 9 January 2018 Accepted: 27 February 2018 Published: 19 March 2018

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Abstract

The pioneer theorem of Weddernburn on commutativity of division rings was proved in the very beginning of twentieth century. Aside from its own intrinsic beauty and important role in many diverse parts of algebra specially, the theorem serves as the starting point for investigations of certain kinds of conditions that render a ring commutative. A large part of the results in this area was developed in the hands of many distinguished mathematicians like Jacobson, Herstein, Kaplansky, Faith, Martindale, Nakayama, Bell and many others. The purpose of the present paper is to investigate commutativity of a ring with unity 1 satisfying certain polynomial constraints. The main result of the first section asserts that a ring is commutative if at least one of the integral exponent used in the polynomial constraints of the theorem is zero and the ring also satisfies the property Q(n) Further, in the second section, commutativity of a ring with unity 1 has also been established under a set of different polynomial identities applying the most frequently used technique known as Streb’s classification. Finally, in the last section, these results of the foregoing sections are further extended to a special class of rings called as one sided s - unital rings.

Published in	International Journal of Data Science and Analysis (Volume 4, Issue 1)
DOI	10.11648/j.ijdsa.20180401.13
Page(s)	13-19
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), 2024. Published by Science Publishing Group

Keywords

Associative Ring, Factor Subring, Polynomial Constraints, Nilpotent Elements, Commutators, Center of Ring, s - Unital Ring and Commutativity

References

[1]	Abujabal, H. A. S.: on commutativity of left s- unital rings, Acta Sci. Math. 56(1992). 51-62.
[2]	Abujabal, H. A. S., Obaid, M. A.: Some commutativity theorems for right s - unital rings, Math. Japonica 37 (1999), 591-600.
[3]	Abujabal, H. A. S. and Ashraf M.: On commutativity of rings involving certain polynomial Constraints, Algebra Colloq. 5(1998), 111-116.
[4]	Ashraf M.: A commutativity theorem for associative rings, Arch. Math (Brno) 31 (1995), 201-204.
[5]	Ashraf M.: Commutativity of rings with certain constraints, Proc. R. Ir. Acad. 96 A, No. 2 (1996), 177-183.
[6]	Bell, H. E., Quadri, M. A. and Ashraf, M.: Commutativity of rings with some commutator constraints, Rad. Mat. 5(1989), 223-230.
[7]	Bell H. E.:A commutativity condition for rings, Canad. J. Math. 28(1979), 986-991.
[8]	Bell H. E.: On power maps and ring commutativity, Canad. Mathematics Bulletin 21(1978), 399-404.
[9]	Bell H. E.: The identity (xy)ⁿ=xⁿyⁿ, does it buy commutativity, Mathematics Mag. 55(1982), 165-170.
[10]	Bell H. E.: Commutativity of rings with constraints on commutators, Resultate Math. 8 (1985), 123-131.
[11]	Harmanci, A.: Two elementary commutativity theorems for rings, Acta. Math. Sci. Hungar 29(1977), 23-29.
[12]	Herstein, I. N.: Two remarks on commutativity of rings, Canadian Journal Math. 9(1957), 583-586.
[13]	Herstein, I. N.: Power maps in rings, Michigan J. 8(1961), 583-586.
[14]	Hirano, Y., Kobayashi, Y. and Tominaga, H: Commutativity theorems for certain rings, Math. J. Okayama Univ. 22(1980), 65-72.
[15]	Hirano, Y., Kobayashi, Y. and Tominaga, H: Some polynomial identities and Commutativity of s - unital rings, Math. J. Okayama Univ. 24(1982), 7-13.
[16]	Hirano, Y., Hongan, H. and Tominaga, H: Commutativity theorems for certain rings, Math. J. Okayama Univ. 22(1980), 65-72.
[17]	Jacobson, N.: Structure theory of algebraic algebras of bounded degree, Ann. of Math. 46(1945), 695-707.
[18]	Jacobson, N.: Structure of rings, Amer. Math. Soc. Colloq. Publ. 37(1964).
[19]	Kezlan, T. P.: A not on commutativity of semi-prime PI rings, Math Japonica 26 (1982), 267-268.
[20]	Komatsu H.: A commutativity theorem for rings, Math. Journal. Okayam Univ. 26(1984), 135-139.
[21]	Komatsu H.: A commutativity theorem for rings-II, Osaka Journal of Math. 22 (1985), 811-814.
[22]	Komatsu H. and Tominaga, H.: Chacron’s conditions and commutativity theorems, Math. Journal Okayam Univ. 31(1989), 101-120.
[23]	Komatsu H. and Tominaga, H.: Some commutativity theorems for s - unital rings, Resultate Math. 15(1989), 335-342.
[24]	Komatsu H. and Tominaga, H.: Some commutativity conditions for rings with unity, Resultate Math. 19(1991), 83-88.
[25]	Komatsu H., Nishinaka, T. and Tominaga, H.: On commutativity of rings, Radovi Mat. 6(1990), 303-311.
[26]	Khan, M. S.;Rings admitting certain decomposition theorems. Journal of Advances in Mathemativs, Vol. 13, No. 01, 12017.
[27]	Nishinaka, T.: A commutativity of rings, Rad. Mat. 6(1990), 357-359.
[28]	Psomopoulos, E.: A commutativity theorem for rings involving a subset of ring, Glasnik Mat. 8(1983), 231-236.
[29]	Psomopoulos, E.: Commutativity theorem for rings and groups, Internat. Journal of Math. and Math. Sci. 7 No.3 (1984), 513-517.
[30]	Psomopoulos, E., Tominaga, H. and Yaqub, A.,: Some commutativity theorems for n- torsion free rings, Math. Journal of Okayama Univ. 23(1981), 37-39.
[31]	Quadri, M. A. and Khan, M. A.: A commutativity theorem for associative ring, Math. Japonica 29(1984), 371-373.
[32]	Raza, M. A., Khan, M. S. and Rehamn, Nadeem-ur; Derivation in prime and semiprime rings with Banach algebras, Italian Journal of Pure and applied Mathematics, Vol. 40, 2018 (To Appear).
[33]	Streb’s, W.: Zur struktur nichtkommutativer ringe, Math. J. Okayama Univ. 31(1989), 135-140.
[34]	Tominaga, H. and Yaqub, A.: Commutativity theorems for rings with constraints involving a commutative subset, Resultate Math. 11(1987), 186-192.
[35]	Tominaga, H.: Some commutativity theorems for s - unital rings satisfying certain constraint, Resultate Math. 6(1983), 217-219.

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Mohammad Shadab Khan. (2018). On Commutativity of Rings Under Certain Polynomial Constraints. International Journal of Data Science and Analysis, 4(1), 13-19. https://doi.org/10.11648/j.ijdsa.20180401.13

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Mohammad Shadab Khan. On Commutativity of Rings Under Certain Polynomial Constraints. Int. J. Data Sci. Anal. 2018, 4(1), 13-19. doi: 10.11648/j.ijdsa.20180401.13

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Mohammad Shadab Khan. On Commutativity of Rings Under Certain Polynomial Constraints. Int J Data Sci Anal. 2018;4(1):13-19. doi: 10.11648/j.ijdsa.20180401.13

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@article{10.11648/j.ijdsa.20180401.13,
  author = {Mohammad Shadab Khan},
  title = {On Commutativity of Rings Under Certain Polynomial Constraints},
  journal = {International Journal of Data Science and Analysis},
  volume = {4},
  number = {1},
  pages = {13-19},
  doi = {10.11648/j.ijdsa.20180401.13},
  url = {https://doi.org/10.11648/j.ijdsa.20180401.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdsa.20180401.13},
  abstract = {The pioneer theorem of Weddernburn on commutativity of division rings was proved in the very beginning of twentieth century. Aside from its own intrinsic beauty and important role in many diverse parts of algebra specially, the theorem serves as the starting point for investigations of certain kinds of conditions that render a ring commutative. A large part of the results in this area was developed in the hands of many distinguished mathematicians like Jacobson, Herstein, Kaplansky, Faith, Martindale, Nakayama, Bell and many others. The purpose of the present paper is to investigate commutativity of a ring with unity 1 satisfying certain polynomial constraints. The main result of the first section asserts that a ring is commutative if at least one of the integral exponent used in the polynomial constraints of the theorem is zero and the ring also satisfies the property Q(n)  Further, in the second section, commutativity of a ring with unity 1 has also been established under a set of different polynomial identities applying the most frequently used technique known as Streb’s classification. Finally, in the last section, these results of the foregoing sections are further extended to a special class of rings called as one sided s - unital rings.},
 year = {2018}
}

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TY  - JOUR
T1  - On Commutativity of Rings Under Certain Polynomial Constraints
AU  - Mohammad Shadab Khan
Y1  - 2018/03/19
PY  - 2018
N1  - https://doi.org/10.11648/j.ijdsa.20180401.13
DO  - 10.11648/j.ijdsa.20180401.13
T2  - International Journal of Data Science and Analysis
JF  - International Journal of Data Science and Analysis
JO  - International Journal of Data Science and Analysis
SP  - 13
EP  - 19
PB  - Science Publishing Group
SN  - 2575-1891
UR  - https://doi.org/10.11648/j.ijdsa.20180401.13
AB  - The pioneer theorem of Weddernburn on commutativity of division rings was proved in the very beginning of twentieth century. Aside from its own intrinsic beauty and important role in many diverse parts of algebra specially, the theorem serves as the starting point for investigations of certain kinds of conditions that render a ring commutative. A large part of the results in this area was developed in the hands of many distinguished mathematicians like Jacobson, Herstein, Kaplansky, Faith, Martindale, Nakayama, Bell and many others. The purpose of the present paper is to investigate commutativity of a ring with unity 1 satisfying certain polynomial constraints. The main result of the first section asserts that a ring is commutative if at least one of the integral exponent used in the polynomial constraints of the theorem is zero and the ring also satisfies the property Q(n)  Further, in the second section, commutativity of a ring with unity 1 has also been established under a set of different polynomial identities applying the most frequently used technique known as Streb’s classification. Finally, in the last section, these results of the foregoing sections are further extended to a special class of rings called as one sided s - unital rings.
VL  - 4
IS  - 1
ER  -

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Author Information

Mohammad Shadab Khan

Department of Commerce, Aligarh Muslim University, Aligarh, India

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APA Style

Mohammad Shadab Khan. (2018). On Commutativity of Rings Under Certain Polynomial Constraints. International Journal of Data Science and Analysis, 4(1), 13-19. https://doi.org/10.11648/j.ijdsa.20180401.13

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ACS Style

Mohammad Shadab Khan. On Commutativity of Rings Under Certain Polynomial Constraints. Int. J. Data Sci. Anal. 2018, 4(1), 13-19. doi: 10.11648/j.ijdsa.20180401.13

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AMA Style

Mohammad Shadab Khan. On Commutativity of Rings Under Certain Polynomial Constraints. Int J Data Sci Anal. 2018;4(1):13-19. doi: 10.11648/j.ijdsa.20180401.13

Copy | Download

@article{10.11648/j.ijdsa.20180401.13,
  author = {Mohammad Shadab Khan},
  title = {On Commutativity of Rings Under Certain Polynomial Constraints},
  journal = {International Journal of Data Science and Analysis},
  volume = {4},
  number = {1},
  pages = {13-19},
  doi = {10.11648/j.ijdsa.20180401.13},
  url = {https://doi.org/10.11648/j.ijdsa.20180401.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdsa.20180401.13},
  abstract = {The pioneer theorem of Weddernburn on commutativity of division rings was proved in the very beginning of twentieth century. Aside from its own intrinsic beauty and important role in many diverse parts of algebra specially, the theorem serves as the starting point for investigations of certain kinds of conditions that render a ring commutative. A large part of the results in this area was developed in the hands of many distinguished mathematicians like Jacobson, Herstein, Kaplansky, Faith, Martindale, Nakayama, Bell and many others. The purpose of the present paper is to investigate commutativity of a ring with unity 1 satisfying certain polynomial constraints. The main result of the first section asserts that a ring is commutative if at least one of the integral exponent used in the polynomial constraints of the theorem is zero and the ring also satisfies the property Q(n)  Further, in the second section, commutativity of a ring with unity 1 has also been established under a set of different polynomial identities applying the most frequently used technique known as Streb’s classification. Finally, in the last section, these results of the foregoing sections are further extended to a special class of rings called as one sided s - unital rings.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - On Commutativity of Rings Under Certain Polynomial Constraints
AU  - Mohammad Shadab Khan
Y1  - 2018/03/19
PY  - 2018
N1  - https://doi.org/10.11648/j.ijdsa.20180401.13
DO  - 10.11648/j.ijdsa.20180401.13
T2  - International Journal of Data Science and Analysis
JF  - International Journal of Data Science and Analysis
JO  - International Journal of Data Science and Analysis
SP  - 13
EP  - 19
PB  - Science Publishing Group
SN  - 2575-1891
UR  - https://doi.org/10.11648/j.ijdsa.20180401.13
AB  - The pioneer theorem of Weddernburn on commutativity of division rings was proved in the very beginning of twentieth century. Aside from its own intrinsic beauty and important role in many diverse parts of algebra specially, the theorem serves as the starting point for investigations of certain kinds of conditions that render a ring commutative. A large part of the results in this area was developed in the hands of many distinguished mathematicians like Jacobson, Herstein, Kaplansky, Faith, Martindale, Nakayama, Bell and many others. The purpose of the present paper is to investigate commutativity of a ring with unity 1 satisfying certain polynomial constraints. The main result of the first section asserts that a ring is commutative if at least one of the integral exponent used in the polynomial constraints of the theorem is zero and the ring also satisfies the property Q(n)  Further, in the second section, commutativity of a ring with unity 1 has also been established under a set of different polynomial identities applying the most frequently used technique known as Streb’s classification. Finally, in the last section, these results of the foregoing sections are further extended to a special class of rings called as one sided s - unital rings.
VL  - 4
IS  - 1
ER  -

Copy | Download