Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero. In this paper we investigate the structures of Zgradings 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 and not isomorphic as graded algebras. We also present the notion of central Z-gradings on E and we show that its Z-graded polynomial identities are closely related to the Z2-graded polynomial identities of Z2-gradings on E.
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