Polynomial functions on a class of finite non-commutative rings

Let [Formula: see text] be a finite non-commutative ring with [Formula: see text]. By a polynomial function on [Formula: see text], we mean a function [Formula: see text] induced by a polynomial [Formula: see text] via right substitution of the variable [Formula: see text], i.e. [Formula: see text] for every [Formula: see text]. In this paper, we study the polynomial functions of the free [Formula: see text]-algebra [Formula: see text], for [Formula: see text], with a basis [Formula: see text] consisting of central elements satisfying [Formula: see text] for every [Formula: see text]. Our investigation revolves around assigning a polynomial [Formula: see text] over [Formula: see text] in non-commuting variables [Formula: see text] and [Formula: see text] to each polynomial [Formula: see text] in [Formula: see text]; and describing the polynomial functions on [Formula: see text] through the polynomial functions induced on [Formula: see text] by polynomials in [Formula: see text] and by their assigned polynomials in the non-commuting variables [Formula: see text] and [Formula: see text]. 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.