View source for Amitsur-Levitzki theorem

A basic result in the theory of polynomial identity rings (PI-rings). A ring $R$ is a PI-ring (cf. also [[PI-algebra]]) if there is a polynomial in the [[free associative algebra]] $\mathbf{Z}\langle x_1,x_2,\ldots\rangle$ which vanishes under all substitutions of ring elements for the variables. The ''standard polynomial'' of degree $n$ is the polynomial
$$
S_n(x_1,\ldots,x_n) = \sum_{\sigma \in \Sigma_n} \mathrm{sign}(\sigma) \, x_{\sigma(1)}\cdots x_{\sigma(n)}
$$
where $\Sigma_n$ is the [[symmetric group]] on $n$ letters. Since $S_2 = x_1 x_2 - x_2 x_1$, a ring is commutative if and only if it satisfies $S_2$ (cf. also [[Commutative ring]]). The Amitsur–Levitzki theorem says that the ring of $(n\times n)$-matrices over a commutative ring satisfies the standard polynomial of degree $2n$, and no polynomial of lower degree.

There are five different published proofs of this theorem (up to 1996). The original proof by S. Amitsur and J. Levitzki [[#References|[a1]]] is a combinatorial argument using matrix units. Because the standard polynomial is linear in each variable, it is enough to prove the theorem when matrix units are substituted for the variables. A streamlined version of this proof can be found in [[#References|[a3]]].

The second proof, by B. Kostant [[#References|[a2]]], depends upon the Frobenius theory of representations of the alternating group. Kostant's paper was also the first to relate the polynomial identities satisfied by matrices with traces, a theme which was later developed by C. Procesi [[#References|[a4]]] and Yu.P. Razmyslov [[#References|[a5]]] and influenced much research. The point is that the trace defines a non-degenerate [[bilinear form]] on $(n\times n)$-matrices.

The third proof, by R.G. Swan [[#References|[a7]]], translates the problem into [[graph theory]]. The underlying arguments are similar to those in the original proof by Amitsur and Levitzki, but the graph-theoretical approach has led to some generalizations, as in [[#References|[a8]]]. The fourth proof, by Razmyslov [[#References|[a5]]], is probably the most natural, in that the theorem is deduced in a direct way from the multilinear form of the [[Cayley–Hamilton theorem]]. The fifth proof, by S. Rosset [[#References|[a6]]], also depends on the Cayley–Hamilton theorem, as well as on elementary properties of the [[exterior algebra]]. It is the shortest and most elegant proof, but is not at all straightforward.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S.A. Amitsur,   J. Levitzki,   "Minimal identities for algebras"  ''Proc. Amer. Math. Soc.'' , '''1'''  (1950)  pp. 449–463</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Kostant,   "A theorem of Frobenius, a theorem of Amitsur–Levitzki, and cohomology theory"  ''J. Math. Mech.'' , '''7'''  (1958)  pp. 237–264</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  D.S. Passman,   "The algebraic structure of group rings" , Wiley  (1977)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Procesi,   "The invariant theory of $n\times n$ matrices"  ''Adv. in Math.'' , '''19'''  (1976)  pp. 306–381</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  Yu.P. Razmyslov,   "Trace identities of full matrix algebras over a field of characteristic zero"  ''Math. USSR Izv.'' , '''8'''  (1974)  pp. 727–760  ''Izv. Akad. Nauk SSSR'' , '''38'''  (1974)  pp. 723–756</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Rosset,   "A new proof of the Amitsur–Levitzki identity"  ''Israel J. Math.'' , '''23'''  (1976)  pp. 187–188</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  R.G. Swan,   "An application of graph theory to algebra"  ''Proc. Amer. Math. Soc.'' , '''14'''  (1963)  pp. 367–380</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  J. Szigeti,   Z. Tuza,   G. Revesz,   "Eulerian polynomial identities on matrix rings"  ''J. Algebra'' , '''161'''  (1993)  pp. 90–101</TD></TR>
</table>

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