# 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:

• 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 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.
Title standard identity StandardIdentity 2013-03-22 14:21:10 2013-03-22 14:21:10 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 16R10