Multilinear Polynomials Are Surjective on Algebras With Surjective Inner Derivations

TL;DR

This paper proves that in unital algebras with surjective inner derivations, any element can be expressed as a nonzero multilinear polynomial evaluated at some algebra elements, highlighting the polynomial's surjectivity.

Contribution

It establishes that multilinear polynomials are surjective on algebras with surjective inner derivations, a novel result connecting polynomial evaluation and algebraic derivation properties.

Findings

Multilinear polynomials are surjective on certain algebras.

Any element in such algebras can be represented as a polynomial evaluation.

The result links polynomial surjectivity to the algebra's derivation structure.

Abstract

Let $f(X_1,\dots, X_n)$ be a nonzero multilinear noncommutative polynomial. If $A$ is a unital algebra with a surjective inner derivation, then every element in $A$ can be written as $f(a_1,\dots,a_n)$ for some $a_i\in A$.