Homolya, Szilvia; Szigeti, Jenő A variant of Rosset’s approach to the Amitsur-Levitzki theorem and some \(\mathbb{Z}_2\)-graded identities of \(\mathrm{M}_n(E)\). (English) Zbl 1517.16020 Turk. J. Math. 46, No. 5, SI-2, 1864-1870 (2022). The celebrated theorem of A. S. Amitsur and J. Levitzki [Proc. Am. Math. Soc. 1, 449–463 (1950; Zbl 0040.01101)] states that the matrix algebra \(M_n(F)\) over a field \(F\) (or more generally over a commutative unital ring) satisfies the standard identity \(s_{2n}\), the latter being the alternating sum of all multilinear monomials of degree \(2n\) in the variables \(x_1, \dots, x_{2n}\). Moreover this is the identity of the least degree for \(M_n(F)\). There are several different proofs of this theorem, the shortest of them probably that of S. Rosset [Isr. J. Math. 23, 187–188 (1976; Zbl 0322.15020)].The full matrix algebras \(M_n(F)\) together with \(M_n(E)\) where \(E\) is the infinite dimensional Grassmann algebra, are of extreme importance in PI theory. In characteristic 0 they, together with the algebras \(M_{a,b}(E)\) generate the T-prime varieties. Recall that \(E=E_0\oplus E_1\) is naturally a superalgebra (a 2-graded algebra), and \(M_{a,b}(E)\) is the subalgebra of all matrices of order \(a+b\) over \(E\) divided into four blocks. The two blocks on the main diagonal have entries coming from \(E_0\) while the off-diagonal blocks have their entries from \(E_1\).By following ideas from the above mentioned paper by Rosset, the authors of the paper under review deduce a new 2-graded polynomial identity for the algebra \(M_n(E)\). Reviewer: Plamen Koshlukov (Campinas) MSC: 16R10 \(T\)-ideals, identities, varieties of associative rings and algebras 16R40 Identities other than those of matrices over commutative rings 16W50 Graded rings and modules (associative rings and algebras) 15A24 Matrix equations and identities 15A75 Exterior algebra, Grassmann algebras 15B33 Matrices over special rings (quaternions, finite fields, etc.) Keywords:full matrix algebra over the infinite dimensional Grassmann algebra; Amitsur-Levitzki theorem on \(n\times n\) matrices Citations:Zbl 0040.01101; Zbl 0322.15020 PDFBibTeX XMLCite \textit{S. Homolya} and \textit{J. Szigeti}, Turk. J. Math. 46, No. 5, 1864--1870 (2022; Zbl 1517.16020) Full Text: DOI References: [1] Aljadeff E, Giambruno A, Procesi C, Regev A. Rings with Polynomial Identities and Finite Dimensional Algebra. AMS Colloquium Publications, Volume 66, American Mathematical Society, Providence, RI, 2020. · Zbl 1471.16001 [2] Amitsur SA. The T-ideals of the free ring. 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