GRADED IDENTITIES AND PI EQUIVALENCE OF ALGEBRAS IN POSITIVE CHARACTERISTIC

ABSTRACT

The algebras M _{a, b} (E) ? E and M _a+b (E) are PI equivalent over a field of characteristic 0 where E is the infinite-dimensional Grassmann algebra. This result is a part of the well-known tensor product theorem. It was first proved by Kemer in 1984–1987 (see Kemer 1991 Kemer , A. ( 1991 ). Ideals of identities of associative algebras . Translations Math. Monographs 87 . Providence , RI : Am. Math. Soc. .Crossref] , Google Scholar]); other proofs of it were given by Regev (1990 Regev , A. ( 1990 ). Tensor products of matrix algebras over the Grassmann algebra . J. Alg. 133 ( 2 ): 512 – 526 . CROSSREF]  Google Scholar]), and in several particular cases, by Di Vincenzo (1992 Di Vincenzo , O. M. ( 1992 ). On the graded identities of M_{1, 1}( E) . Israel J. Math. 80 ( 3 ): 323 – 335 .Crossref], Web of Science ®] , Google Scholar]), and by the authors (2004). Using graded polynomial identities, we obtain a new elementary proof of this fact and show that it fails for the T-ideals of the algebras M _{1, 1}(E) ? E and M ₂(E) when the base field is infinite and of characteristic p > 2. The algebra M _{a, a} (E) ? E satisfies certain graded identities that are not satisfied by M _2a (E). In another paper we proved that the algebras M _{1, 1}(E) and E ? E are not PI equivalent in positive characteristic, while they do satisfy the same multilinear identities.