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

Year 2022, , 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

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