2024/09/16 by Amr Ali Abdulkader Al-Maktry, Susan F. El-Deken, Al-Maktry, Amr Ali Abdulkader +1
Mathematics · #math.RA #math.CO
paper · pdf · doi:10.48550/arxiv.2409.10208
Let R be a finite non-commutative ring with 1≠ 0. By a polynomial function on R, we mean a function F\colon R\longrightarrow R induced by a polynomial f=∑i=0naixi∈ R[x] via right substitution of the variable x, i.e. F(a)=f(a)= ∑i=0naiai for every a∈ R. In this paper, we study the polynomial functions of the free R-algebra with a central basis \1,β1,…,βk\ (k≥ 1) such that βiβj=0 for every 1≤ i,j≤ k, R[β1,…,βk]. %, the ring of dual numbers over R in k variables. Our investigation revolves around assigning a polynomial λf(y,z) over R in non-commutating variables y and z to each polynomial f in R[x]; and describing the polynomial functions on R[β1,…,βk] through the polynomial functions induced on R by polynomials in R[x] and by their assigned polynomials in the non-commutating variables y and z. %and analyzing the resulting polynomial functions on R[β1,…,βk]. By extending results from the commutative case to the non-commutative scenario, we demonstrate that several properties and theorems in the commutative case can be generalized to the non-commutative setting with appropriate adjustments.