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

Year 2022, Volume: 31 Issue: 31, 1 - 12, 17.01.2022

Amr Ali Abdulkader Al-maktry

https://doi.org/10.24330/ieja.1058380

Abstract

Let $R$ be a commutative ring with unity $1\ne 0$. In this paper we introduce the definition of the first derivative property on the ideals of the polynomial ring $R\left[x\right]$. In particular, when $R$ is a finite local ring with principal maximal ideal $m\ne \left\{0\right\}$ of index of nilpotency $e$, where $1<e\le |R/m|+1$, we show that the null ideal consisting of polynomials inducing the zero function on $R$ satisfies this property. As an application, when $R$ is a finite local ring with null ideal satisfying this property, we prove that the stabilizer group of $R$ in the group of polynomial permutations on the ring $R\left[x\right]/\left(x^2\right)$, is isomorphic to a certain factor group of the null ideal.

Keywords

Commutative rings , polynomial ring , null ideal , null polynomial , Henselian ring , finite local ring , dual numbers , polynomial permutation , permutation polynomial , finite permutation group

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.