Residually finite dimensional algebras and polynomial almost identities

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 (respectively, locally nilpotent) ideal of finite codimension if char k = 0 (respectively, 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 is an element of k < x(1), ... ,x(n)> has degree d, and A is a finite k-algebra such that the probability that Q(a(1), ... ,a(n)) = 0 (where alpha(i) is an element of 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,