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 $R\u27e8X\u27e9$, denoted by $[{x}_{1},\mathrm{\dots}{x}_{n}]$, is the polynomial^{}
$$\sum _{\pi}\mathrm{sign}(\pi ){x}_{\pi (1)}\mathrm{\cdots}{x}_{\pi (n)},\text{where}\pi \in {S}_{n}.$$ 
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 $[{x}_{1},\mathrm{\dots}{x}_{n}]$ are that it is multilinear over $R$, and it is alternating, in the sense that $[{r}_{1},\mathrm{\dots}{r}_{n}]=0$ 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.
Title  standard identity 

Canonical name  StandardIdentity 
Date of creation  20130322 14:21:10 
Last modified on  20130322 14:21:10 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16R10 