Almost PI algebras are PI

TL;DR

The paper introduces the concept of almost polynomial identities in associative, Lie, and Jordan algebras, proving that their existence guarantees actual polynomial identities, with additional quantitative results for simple and semisimple cases.

Contribution

It establishes that almost polynomial identities imply genuine polynomial identities across various algebra types, extending the understanding of algebraic identities.

Findings

Almost polynomial identities lead to actual polynomial identities.

Results apply to associative, Lie, and Jordan algebras.

Quantitative results are provided for simple and semisimple algebras.

Abstract

We define the notion of an almost polynomial identity of an associative algebra $R$, and show that its existence implies the existence of an actual polynomial identity of $R$. A similar result is also obtained for Lie algebras and Jordan algebras. We also prove related quantitative results for simple and semisimple algebras.