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

Abstract

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.