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.