In this paper we study tensor products of T -prime T -ideals over infinite fields. The behaviour of these tensor products over a field of characteristic 0 was described by Kemer. First we show, using methods due to Regev, that such a description holds if one restricts oneself to multilinear polynomials only. Second, applying graded polynomial identities, we prove that the tensor product theorem fails for the T -ideals of the algebras M 1,1 (E) and E ⊗ E where E is the infinite-dimensional Grassmann algebra; M 1,1 (E) consists of the 2 × 2 matrices over E having even (i.e., central) elements of E, and the other diagonal consisting of odd (anticommuting) elements of E. Note that these proofs do not depend on the structure theory of T -ideals but are "elementary" ones. All this comes to show once more that the structure theory of T -ideals is essentially about the multilinear polynomial identities.
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