We show that a Cramér–Wold device holds for infinite divisibility of Z^d-valued distributions, i.e. that the distribution of a Z^d-valued random vector X is infinitely divisible if and only if the distribution of a^T X is infinitely divisible for all a ∈ R^d, and that this in turn is equivalent to infinite divisibility of the distribution of a^T X for all a ∈ N^d0 . A key tool for proving this is a Lévy–Khintchine type representation with a signed Lévy measure for the^d.