Let RR be a commutative ring with unity 1≠01≠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.
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
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.
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.