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

Authors : Amr Ali Abdulkader ALMAKTRY

Pages : 1-12

Doi:10.24330/ieja.1058380

View : 18 | Download : 18

Publication Date : 2022-01-17

Article Type : Research Paper

Abstract :Let R R be a commutative ring with unity 1 ≠ 0 1≠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 R R is a finite local ring with principal maximal ideal m ≠ { 0 } m≠{0} of index of nilpotency e e , where 1 < e ≤ | R / m | + 1 1≤e≤|R/m|+1 , we show that the null ideal consisting of polynomials inducing the zero function on R R satisfies this property. As an application, when R R is a finite local ring with null ideal satisfying this property, we prove that the stabilizer group of R R in the group of polynomial permutations on the ring R [ x ] / insert ignore into journalissuearticles values( x 2 ); R[x]/insert ignore into journalissuearticles values(x2); , 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