Polynomial functions on a class of finite non-commutative rings

Let $R$ be a finite non-commutative ring with $1\mathrm{\ne }0$. By a polynomial function on $R$, we mean a function $F\text{\hspace{0.17em}⁣}:R⟶R$ induced by a polynomial $f={\sum }_{i=0}^n{a}_i{x}^i\in R[x]$ via right substitution of the variable $x$, i.e. $F(a)=f(a)={\sum }_{i=0}^n{a}_i{a}^i$ for every $a\in R$. In this paper, we study the polynomial functions of the free $R$-algebra with a central basis $\{1,{\beta }_1,\dots ,{\beta }_k\}$ ($k\ge 1$) such that ${\beta }_i{\beta }_j=0$ for every $1\le i,j\le k$, $R[{\beta }_1,\dots ,{\beta }_k]$. %, the ring of dual numbers over $R$ in $k$ variables. Our investigation revolves around assigning a polynomial ${\lambda }_f(y,z)$ over $R$ in non-commutating variables $y$ and $z$ to each polynomial $f$ in $R[x]$; and describing the polynomial functions on $R[{\beta }_1,\dots ,{\beta }_k]$ 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 $R[{\beta }_1,\dots ,{\beta }_k]$. 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.