standard identity

Let $R$ be a commutative ring and $X$ be a set of non-commuting variables over $R$ . The standard identity of degree $n$ in $R\langle X\rangle$ , denoted by $[x_{1},\ldots\,x_{n}]$ , is the polynomial

$\sum_{\pi}\operatorname{sign}(\pi)x_{\pi(1)}\cdots x_{\pi(n)},\mbox{ where }% \pi\in S_{n}.$

Remarks:

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A ring $R$ satisfying the standard identity of degree 2 (i.e., $[R,R]=0$ ) is commutative. In this sense, algebras satisfying a standard identity is a generalization of the class of commutative algebras.
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Two immediate properties of $[x_{1},\ldots\,x_{n}]$ are that it is multilinear over $R$ , and it is alternating, in the sense that $[r_{1},\ldots\,r_{n}]=0$ whenever two of the $r_{i}^{\prime}s$ are equal. Because of these two properties, one can show that an n-dimensional algebra $R$ over a field $k$ is a PI-algebra, satisfying the standard identity of degree $n+1$ . As a corollary, $\mathbb{M}_{n}(k)$ , the $n\times n$ matrix ring over a field $k$ , is a PI-algebra satisfying the standard identity of degree $n^{2}+1$ . In fact, Amitsur and Levitski have shown that $\mathbb{M}_{n}(k)$ actually satisfies the standard identity of degree $2n$ .

References

1 S. A. Amitsur and J. Levitski, Minimal identities for algebras, Proc. Amer. Math. Soc., 1 (1950) 449-463.