Residually finite dimensional algebras and polynomial almost identities

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 $Q\in k⟨x_1,\dots ,x_n⟩$ has degree $d$, and $A$ is a finite $k$-algebra such that the probability that $Q(a_1,\dots ,a_n)=0$ (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.