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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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##### References:
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