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

Let R be a commutative ring with unity 1 ≠ 0. In this paper we introduce the definition of the first derivative property on the ideals of the polynomial ring R[x]. In particular, when R is a finite local ring with principal maximal ideal m ≠ {0} of index of nilpotency e, where 1 < e ≤ |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[x]/(x ²), is isomorphic to a certain factor group of the null ideal.