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On a property of the ideals of the polynomial ring $R[x]$

Year 2022, Volume: 31 Issue: 31, 1 - 12, 17.01.2022
https://doi.org/10.24330/ieja.1058380

Abstract

Let RR be a commutative ring with unity 101≠0. In this paper we introduce the definition of the first derivative property on the ideals of the polynomial ring R[x]R[x]. In particular, when RR is a finite local ring with principal maximal ideal m{0}m≠{0} of index of nilpotency ee, where 1<e|R/m|+11≤e≤|R/m|+1, we show that the null ideal consisting of polynomials inducing the zero function on RR satisfies this property. As an application, when RR is a finite local ring with null ideal satisfying this property, we prove that the stabilizer group of RR in the group of polynomial permutations on the ring R[x]/(x2)R[x]/(x2), is isomorphic to a certain factor group of the null ideal.

References

  • H. Al-Ezeh, A. A. Al-Maktry and S. Frisch, Polynomial functions on rings of dual numbers over residue class rings of the integers, Math. Slovaca, 71(5) (2021), 1063-1088.
  • B. R. McDonald, Finite rings with identity, Pure and Applied Mathematics, Vol. 28, Marcel Dekker, Inc., New York, 1974.
  • A. A. Necaev, Polynomial transformations of finite commutative local rings of principal ideals, Math. Notes, 27(5-6) (1980), 425-432. translate from Mat. Zametki, 27(6) (1980), 885-897.
  • W. Nobauer, Uber die Ableitungen der Vollideale, Math. Z., 75 (1961), 14-21.
  • M. W. Rogers and C. Wickham, Polynomials inducing the zero function on local rings, Int. Electron. J. Algebra, 22 (2017), 170-186.
Year 2022, Volume: 31 Issue: 31, 1 - 12, 17.01.2022
https://doi.org/10.24330/ieja.1058380

Abstract

References

  • H. Al-Ezeh, A. A. Al-Maktry and S. Frisch, Polynomial functions on rings of dual numbers over residue class rings of the integers, Math. Slovaca, 71(5) (2021), 1063-1088.
  • B. R. McDonald, Finite rings with identity, Pure and Applied Mathematics, Vol. 28, Marcel Dekker, Inc., New York, 1974.
  • A. A. Necaev, Polynomial transformations of finite commutative local rings of principal ideals, Math. Notes, 27(5-6) (1980), 425-432. translate from Mat. Zametki, 27(6) (1980), 885-897.
  • W. Nobauer, Uber die Ableitungen der Vollideale, Math. Z., 75 (1961), 14-21.
  • M. W. Rogers and C. Wickham, Polynomials inducing the zero function on local rings, Int. Electron. J. Algebra, 22 (2017), 170-186.
There are 5 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Amr Ali Abdulkader Al-maktry This is me

Publication Date January 17, 2022
Published in Issue Year 2022 Volume: 31 Issue: 31

Cite

APA Al-maktry, A. A. A. (2022). On a property of the ideals of the polynomial ring $R[x]$. International Electronic Journal of Algebra, 31(31), 1-12. https://doi.org/10.24330/ieja.1058380
AMA Al-maktry AAA. On a property of the ideals of the polynomial ring $R[x]$. IEJA. January 2022;31(31):1-12. doi:10.24330/ieja.1058380
Chicago Al-maktry, Amr Ali Abdulkader. “On a Property of the Ideals of the Polynomial Ring $R[x]$”. International Electronic Journal of Algebra 31, no. 31 (January 2022): 1-12. https://doi.org/10.24330/ieja.1058380.
EndNote Al-maktry AAA (January 1, 2022) On a property of the ideals of the polynomial ring $R[x]$. International Electronic Journal of Algebra 31 31 1–12.
IEEE A. A. A. Al-maktry, “On a property of the ideals of the polynomial ring $R[x]$”, IEJA, vol. 31, no. 31, pp. 1–12, 2022, doi: 10.24330/ieja.1058380.
ISNAD Al-maktry, Amr Ali Abdulkader. “On a Property of the Ideals of the Polynomial Ring $R[x]$”. International Electronic Journal of Algebra 31/31 (January 2022), 1-12. https://doi.org/10.24330/ieja.1058380.
JAMA Al-maktry AAA. On a property of the ideals of the polynomial ring $R[x]$. IEJA. 2022;31:1–12.
MLA Al-maktry, Amr Ali Abdulkader. “On a Property of the Ideals of the Polynomial Ring $R[x]$”. International Electronic Journal of Algebra, vol. 31, no. 31, 2022, pp. 1-12, doi:10.24330/ieja.1058380.
Vancouver Al-maktry AAA. On a property of the ideals of the polynomial ring $R[x]$. IEJA. 2022;31(31):1-12.