Let R be a finite commutative ring with 1≠0. The set F(R) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units F(R)× is just the set of all unit-valued polynomial functions, that is the set of polynomial functions which map R into its group of units. We show that PR(R[x]/(x2)) the group of polynomial permutations on the ring R[x]/(x2), consisting of permutations represented by polynomials over R, is embedded in a semidirect product of F(R)× by P(R) the group of polynomial permutations on R. In particular, when R=Fq, we prove that PFq(Fq[x]/(x2))≅P(Fq)⋉θF(Fq)×. Furthermore, we count unit-valued polynomial functions (modpn) and obtain canonical representations for these functions.
|Number of pages||12|
|Journal||Applicable Algebra in Engineering, Communication and Computing|
|Publication status||Accepted/In press - 5 Apr 2021|