Graded polynomial identities play an important role in the structure theory of PI algebras.Many properties of the ideals of identities are described in the language of graded identities and graded algebras. In this paper we study the elementary gradings on the algebra UT_n(K) of n × n upper triangular matrices over an infinite field. We describe these gradings by means of the graded identities that they satisfy. Namely we prove that there exist |G|^{n−1} nonisomorphic elementary gradings on UT_n(K) by the finite group G, and show that nonisomorphic gradings produce different graded identities. Furthermore we describe generators for the ideals of graded identities for a given (but arbitrary) elementary grading on UT_n(K), and produce linear bases of the corresponding relatively free graded algebra

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We exhibit minimal bases of the polynomial identities for the matrix algebra M 2 (K) of order two over an infinite field K of characteristic p / = 2. We show that when p = 3 the T -ideal of this algebra is generated by three independent identities, and when p > 3 one needs only two identities: the standard identity of degree four and the Hall identity. Note that the same holds when the base field is of characteristic 0. Furthermore, using the exact form of the basis of the identities for M 2 (K) we give finite minimal set of generators of the T -space of the central polynomials for the algebra M 2 (K). The set of generators depends on the characteristic of the field as well.

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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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