On the group of unit-valued polynomial functions

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