Residually finite dimensional algebras and polynomial almost identities

TL;DR

This paper investigates residually finite dimensional algebras over various fields, establishing connections between almost identities, probabilistic identities, and structural properties like nilpotency and finite codimension ideals.

Contribution

It introduces new results linking almost identities and probabilistic identities to algebraic structure, extending known theorems to broader classes of algebras.

Findings

Residually finite dimensional algebras with almost identities have finite codimension ideals satisfying those identities.

In finite fields, probabilistic identities imply coset identities and finite index ideal identities.

A probabilistic bound on polynomial identities in finite algebras guarantees the polynomial is an actual identity.

Abstract

Let $A$ be a residually finite dimensional algebra (not necessarily associative) over a field $k$. Suppose first that $k$ is algebraically closed. We show that if $A$ satisfies a homogeneous almost identity $Q$, then $A$ has an ideal of finite codimension satisfying the identity $Q$. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra $L$ over $k$ is almost $d$-Engel, then $L$ has a nilpotent (resp. locally nilpotent) ideal of finite codimension if char $k=0$ (resp. char $k > 0$). Next, suppose that $k$ is finite (so $A$ is residually finite). We prove that, if $A$ satisfies a homogeneous probabilistic identity $Q$, then $Q$ is a coset identity of $A$. Moreover, if $Q$ is multilinear, then $Q$ is an identity of some finite index ideal of $A$. Along the way we show that, if $Q\in k\langle x_1,\ldots,x_n\rangle$ has degree $d$, and $A$ is a finite $k$-algebra such that the probability that $Q(a_1, \ldots , a_n)=0$ (where $a_i \in A$ are randomly chosen) is at least $1-2^{-d}$, then $Q$ is an identity of $A$. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon.