Formula omitted]-gradings of full support on the Grassmann algebra

Abstract

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero. In this paper we investigate the structures of Z-gradings on E of full support. Using methods of elementary number theory, we describe the Z-graded polynomial identities for the so-called 2-induced Z-gradings on E of full support. As a consequence of this fact we provide examples of Z-gradings on E which are PI-equivalent but not Z-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent, but non-isomorphic as graded algebras. We also present the notion of central Z-gradings on E and we show that their Z-graded polynomial identities are closely related to the Z2-graded polynomial identities of E.