\(\mathbb{Z}_{k+l}\times\mathbb{Z}_2\)-graded polynomial identities for \(M_{k,l}(E)\otimes E\). (English) Zbl 1147.16302

Summary: Let \(K\) be a field of characteristic zero, and \(E\) be the Grassmann algebra over an infinite-dimensional \(K\)-vector space. We endow \(M_{k,l}(E)\otimes E\) with a \(\mathbb{Z}_{k+l}\times\mathbb{Z}_2\)-grading, and determine a generating set for the ideal of its graded polynomial identities. As a consequence, we prove that \(M_{k,l}(E)\times E\) and \(M_{k+l}(E)\) are PI-equivalent with respect to this grading. In particular, this leads to their ordinary PI-equivalence, a classical result obtained by Kemer.


16R50 Other kinds of identities (generalized polynomial, rational, involution)
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
15A75 Exterior algebra, Grassmann algebras
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[1] A. BERELE, Supertraces and matrices over Grassmann algebras, Advances in Math., 108 (1) (1994), pp. 77-90. Zbl0819.16023 MR1293582 · Zbl 0819.16023
[2] O. M. DI VINCENZO - V. NARDOZZA, Graded polynomial identities for tensor products by the Grassmann Algebra, Comm. Algebra (2002) (in press). Zbl1039.16023 · Zbl 1039.16023
[3] A. R. KEMER, Varieties and Z2-graded algebras, Math. USSR Izv., 25 (1985), pp. 359-374. Zbl0586.16010 MR764308 · Zbl 0586.16010
[4] A. R. KEMER, Ideals of identities of associative algebras, AMS Trans. of Math. Monographs, 87 (1991). Zbl0732.16001 MR1108620 · Zbl 0732.16001
[5] A. REGEV, Tensor product of matrix algebras over the Grassmann algebra, J. Algebra, 133 (1990), pp. 351-369. Zbl0639.16010 MR1067423 · Zbl 0639.16010
[6] S. Y. VASILOVSKY, Zn-graded polynomial identities of the full matrix algebra of order n, Proc. Amer. Math. Soc., 127 (12) (1999), pp. 3517-3524. Zbl0935.16012 MR1616581 · Zbl 0935.16012
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