On the structural behavior of images of polynomials

TL;DR

This paper explores the structure of images of noncommutative polynomials in associative algebras, revealing conditions under which these images generate the entire algebra and analyzing their additive and multiplicative properties.

Contribution

It introduces decomposable polynomials, characterizes their images, and proves that noncommutative infinite simple algebras are generated by polynomial commutators.

Findings

Finite sums of polynomial products contain a nonzero ideal in associative algebras.

For simple algebras, the subring generated by polynomial images is the whole algebra, with minor exceptions.

Noncommutative infinite simple algebras are generated by polynomial commutators.

Abstract

The study of images of noncommutative polynomials on algebras has attracted considerable attention. We investigate polynomial images and the additive structures they generate in associative algebras, focusing on sums and products of values. Motivated by results on additive commutators, we show that finite sums of such products on a nonzero ideal must contains a nonzero ideal, with only minor exceptions. Consequently, for a simple algebra, the subring generated by the image of a noncentral polynomial coincides with the whole algebra, up to a small exceptional case. We further study representations of elements as sums of products of polynomial values, and examine products of additive commutators for matrices over division rings. To simplify multilinear polynomials, we introduce decomposable polynomials and show that, in many cases, their images equal the whole algebra. Finally, we consider polynomial commutators and prove that every noncommutative infinite simple algebra is generated by such elements, together with results on multiplicative commutators, including a complete description for real quaternions.