Polynomial functions on a class of finite non-commutative rings

Let $R$ be a finite non-commutative with $1\ne 0$. By a polynomial function on $R$, we mean a function $F\colon R\longrightarrow R$ induced by a polynomial $f=\sum\limits_{i=0}^{n}a_ix^i\in R[x]$ via right substitution of the variable $x$, i.e.
$F(a)=f(a)= \sum\limits_{i=0}^{n}a_ir^i$ for every $a\in R$. In this paper, we study the polynomial functions of the free $R$-algebra
with a central basis $\{1,\beta_1,\ldots,\beta_k\}$ ($k\ge 1$) such that $\beta_i\beta_j=0$ for every $1\le i,j\le k$, $R[\beta_1,\ldots,\beta_k]$, 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-commutative variables $y$ and $z$ to each polynomial $f$ in $R[x]$; and analyzing the resulting polynomial functions on $R[\beta_1,\ldots,\beta_k]$. 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.