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,M-1 (E) and E circle times E where E is the infinite-dimensional Grassmann algebra; M-1.1(E) consists of the 2 x 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. (C) 2004 Elsevier Inc. All rights reserved.2762836845