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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Let [Formula: see text] be an infinite field of characteristic different from 2, and let [Formula: see text] be the Grassmann algebra of a countable of dimensional [Formula: see text]-vector space [Formula: see text]. In this paper, we study the graded central polynomials of gradings on [Formula: see text] by the groups [Formula: see text] and [Formula: see text], where the basis of the vector space [Formula: see text] is homogeneous. More specifically, we provide a basis for the [Formula: see text]-space of graded central polynomials for [Formula: see text], where the group [Formula: see text] is [Formula: see text] and [Formula: see text].

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