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On the theorem of Amitsur-Levitzki. (English) Zbl 1333.16025
The author gives one more proof of the theorem of Amitsur-Levitzki, which is a cornerstone of the theory of polynomial identities. It states that the algebra of \(n\times n\) matrices over any commutative ring satisfies the standard identity: \[ S_{2n}(x_1,\ldots,x_{2n})=\sum_{\sigma\in S_{2n}}\varepsilon_\sigma x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(2n)}\equiv 0, \] where \(S_{2n}\) is the symmetric group of permutations of \(2n\) elements and \(\varepsilon_\sigma\) denotes the sign of the permutation \(\sigma\in S_{2n}\). There are known several proofs. The present proof shows that the Amitsur-Levitzki theorem is the Cayley-Hamilton identity for the generic Grassmann matrix. This approach is close to the proof of S. Rosset [Isr. J. Math. 23, 187-188 (1976; Zbl 0322.15020)].

MSC:
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
17B01 Identities, free Lie (super)algebras
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