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 $R⟨X⟩$, 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:

• 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 $[x_1,\mathrm{\dots }{x}_n]$ 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 n-dimensional algebra $R$ over a field $k$ is a PI-algebra, satisfying the standard identity of degree $n+1$. As a corollary, ${𝕄}_n(k)$, 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 ${𝕄}_n(k)$ actually satisfies the standard identity of degree $2n$.