On the injectivity of evaluation maps induced by polynomials on certain algebras

TL;DR

This paper investigates when evaluation maps induced by polynomials on associative algebras are injective, concluding that injectivity only occurs for linear polynomials in a single variable.

Contribution

It establishes a clear criterion for the injectivity of polynomial evaluation maps on associative algebras, showing it only holds in the linear, single-variable case.

Findings

Injectivity only for linear polynomials in one variable

Injectivity impossible for polynomials in two or more variables

Provides a complete characterization of injectivity conditions

Abstract

We explore the injectivity of the evaluation map eva f,A from Am A to A, where A is an associative algebra over a field F, and f is a polynomial in m \ge 1 variables with coefficients in F. Our investigation reveals that injectivity is possible only when m equal 1 and f has degree one; for functions in two or more variables, such injectivity is impossible.