Polynomial functions on a class of finite non-commutative rings

TL;DR

This paper explores polynomial functions on a specific class of finite non-commutative rings, extending known results from commutative rings to the non-commutative case with new algebraic insights.

Contribution

It introduces a framework for understanding polynomial functions on non-commutative rings with a central basis, generalizing classical commutative results to non-commutative algebra.

Findings

Polynomial functions on the ring are characterized via associated polynomials in non-commuting variables.

Extension of classical properties of polynomial functions from commutative to non-commutative rings.

Describes the structure of polynomial functions on rings with nilpotent elements in the basis.

Abstract

Let $R$ be a finite non-commutative ring 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. $F(a)=f(a)= \sum\limits_{i=0}^{n}a_ia^i$ 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-commutating variables $y$ and $z$ to each polynomial $f$ in $R[x]$; and describing the polynomial functions on $R[\beta_1,\ldots,\beta_k]$ 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 $R[\beta_1,\ldots,\beta_k]$. 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.