论文标题
残留有限的维代数和多项式身份
Residually finite dimensional algebras and polynomial almost identities
论文作者
论文摘要
让$ a $为剩余的维数代数（不一定是关联），对字段$ k $。首先假设$ k $是代数关闭的。我们表明，如果$ a $满足均质的几乎身份$ q $，那么$ a $具有有限的编辑理想，可以满足身份$ q $。使用Zelmanov的众所周知的结果，我们得出的结论是，如果剩余有限的尺寸lie代数$ l $ $ k $上的$ k $几乎是$ d $ - ENGEL，则$ l $将具有nilpotent（本地nilpotent）的有限编码理想，如果char $ k = 0 $ k = 0 $ char $ k> 0 $ k> $ k> $ k> $ k> 0 $ $）。接下来，假设$ k $是有限的（因此$ a $是残留有限的）。我们证明，如果$ a $满足同质概率身份$ q $，则$ q $是$ a $的coset身份。此外，如果$ q $是多线性的，则$ q $是某些有限索引$ a $的标识。 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 $1-2^{-d}$,然后$ q $是$ a $的标识。这解决了迪克森（Dixon）提出的（仍开放的）群体理论问题的环理论类似物。
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.
