Abstract
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.
Original language | English |
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Pages (from-to) | 1-12 |
Number of pages | 12 |
Journal | International Electronic Journal of Algebra |
Volume | 31 |
Issue number | 31 |
DOIs | |
Publication status | Published - 17 Jan 2022 |
Keywords
- Commutative rings
- dual numbers
- finite local ring
- finite permutation group
- Henselian ring
- null ideal
- null polynomial
- permutation polynomial
- polynomial permutation
- polynomial ring
ASJC Scopus subject areas
- Algebra and Number Theory