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Homestandard identity
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standard identity
Let $R$ be a commutative ring and $X$ be a set of noncommuting 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:

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.

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 ndimensional algebra $R$ over a field $k$ is a PIalgebra, 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 PIalgebra 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) 449463.
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