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 k(x1,..,xn) has degree d, and A is a finite k-Algebra such that the probability that Q(a1,..,an) = 0 (where ai A are randomly chosen) is at least 1-2-d, then Q is an identity of A.