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. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

中文翻译：

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剩余有限维代数和多项式几乎恒等式

让一种是域上的剩余有限维代数（不一定是关联的）ķ. 首先假设ķ是代数闭的。我们证明如果一种满足同质几乎恒等式问， 然后一种有一个满足恒等式的有限余维理想问. 使用 Zelmanov 的著名结果，我们得出结论，如果剩余有限维李代数大号超过ķ差不多d——恩格尔，然后大号具有有限余维的幂零（分别为局部幂零）理想，如果ķ = 0（分别，字符ķ > 0）。接下来，假设ķ是有限的（所以一种是剩余有限的）。我们证明，如果一种满足同质概率恒等式问， 然后问是一个陪集标识一种. 此外，如果问是多线性的，那么问是某个有限指数理想的恒等式一种. 一路走来，我们证明如果问 ∈ ķ〈X1,…,Xn〉有学位d， 和一种是一个有限的ķ-代数使得概率问(一种1,…,一种n) = 0（在哪里一种一世 ∈ 一种是随机选择的）至少是1 - 2-d， 然后问是一个身份一种. 这解决了 Dixon 提出的（仍然开放的）群论问题的环论类似物，