GRADED IDENTITIES AND PI EQUIVALENCE OF ALGEBRAS IN POSITIVE CHARACTERISTIC#.

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); other proofs of it were given by Regev (1990), and in several particular cases, by Di Vincenzo (1992), 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 2 a ( 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. [ABSTRACT FROM AUTHOR]