## 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