# 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 [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 [a3].

The second proof, by B. Kostant [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 [a4] and Yu.P. Razmyslov [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 [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 [a8]. The fourth proof, by Razmyslov [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 [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

[a1] | S.A. Amitsur, J. Levitzki, "Minimal identities for algebras" Proc. Amer. Math. Soc. , 1 (1950) pp. 449–463 |

[a2] | B. Kostant, "A theorem of Frobenius, a theorem of Amitsur–Levitzki, and cohomology theory" J. Math. Mech. , 7 (1958) pp. 237–264 |

[a3] | D.S. Passman, "The algebraic structure of group rings" , Wiley (1977) |

[a4] | C. Procesi, "The invariant theory of $n\times n$ matrices" Adv. in Math. , 19 (1976) pp. 306–381 |

[a5] | 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 |

[a6] | S. Rosset, "A new proof of the Amitsur–Levitzki identity" Israel J. Math. , 23 (1976) pp. 187–188 |

[a7] | R.G. Swan, "An application of graph theory to algebra" Proc. Amer. Math. Soc. , 14 (1963) pp. 367–380 |

[a8] | J. Szigeti, Z. Tuza, G. Revesz, "Eulerian polynomial identities on matrix rings" J. Algebra , 161 (1993) pp. 90–101 |

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Amitsur–Levitzki theorem.

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