$\mathbb{Z}$-gradings of full support on the Grassmann algebra

Let $E$ be the infinite dimensional Grassmann algebra over a field $F$ of characteristic zero. In this paper we investigate the structures of $\mathbb{Z}$-gradings on $E$ of full support. Using methods of elementary number theory, we describe the $\mathbb{Z}$-graded polynomial identities for the so-called $2$-induced $\mathbb{Z}$-gradings on $E$ of full support. As a consequence of this fact we provide examples of $\mathbb{Z}$-gradings on $E$ which are PI-equivalent but not $\mathbb{Z}$-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 $\mathbb{Z}$-gradings on $E$ and we show that its $\mathbb{Z}$-graded polynomial identities are closely related to the $\mathbb{Z}_{2}$-graded polynomial identities of $\mathbb{Z}_{2}$-gradings on $E$.