Graded identities for algebras with elementary gradings over an infinite field

Abstract

AbstractLet $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$-grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$, $b\in \mathbb {N}$, we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$, as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$-algebras which are not PI equivalent. Actually, we prove that the $T_{G}$-ideal of the former algebra is contained in the $T$-ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.