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

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