Let R be a finite commutative ring. The set \({{\mathcal{F}}}(R)\) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units \({{\mathcal{F}}}(R)^\times \) is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on \(R[x]/(x^2)=R[\alpha ]\), the ring of dual numbers over R, and show that the group \({\mathcal{P}}_{R}(R[\alpha ])\), consisting of those polynomial permutations of \(R[\alpha ]\) represented by polynomials in R[x], is embedded in a semidirect product of \({{\mathcal{F}}}(R)^\times \) by the group \({\mathcal{P}}(R)\) of polynomial permutations on R. In particular, when \(R={\mathbb{F}}_q\), we prove that \({\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times \). Furthermore, we count unit-valued polynomial functions on the ring of integers modulo \({p^n}\) and obtain canonical representations for these functions.

令R为有限交换环。该组\（{{\ mathcal {F}}}（R）\）的多项式函数[R是逐点的操作的有限交换环。它的单位组\({{\mathcal{F}}}(R)^\times \)只是所有单位值多项式函数的集合。我们探讨多项式置换\（R [X] /（X ^ 2）= R [\α-\），用双数的环- [R，并显示该组\（{\ mathcal {P}} _ { R}(R[\alpha ])\)，由R [ x ] 中的多项式表示的\(R[\alpha ]\) 的多项式排列组成，嵌入在\（{{\ mathcal {F}}}（R）^ \倍\）由组\（{\ mathcal {P}}（R）\）上多项式排列ř。特别地，当\(R={\mathbb{F}}_q\) 时，我们证明\({\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}} _q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^ \ times \）。此外，我们计算整数环上的单位值多项式函数，取模\({p^n}\)并获得这些函数的规范表示。