Maximum number of solutions of $x^q = x$ in a finite noncommutative algebra

TL;DR

This paper establishes an upper bound on the solutions of the equation x^q = x in finite noncommutative algebras over fields with q elements and characterizes rings where this bound is achieved.

Contribution

It provides a bound on the number of solutions and a complete characterization of rings attaining this maximum in finite noncommutative algebras.

Findings

Derived an explicit upper bound for solutions in such algebras.

Characterized rings that reach the maximum number of solutions.

Enhanced understanding of solution structure in noncommutative algebraic systems.

Abstract

We obtain a bound on the number of solutions of $x^q=x$ in a finite noncommutative algebra over a field with $q$ elements. Furthermore, we completely characterize those rings for which this maximum number is attained.