I am studying the fast Fourier transform algorithm in von zur Gathen and Gerard's Modern computer algebra. In it (p. 220), they seem to claim that in a ring with unity where we can apply the discrete Fourier transform, i.e; a ring with a primitive $n$-th root of unity, we also have an inverse. But this is only the case if the element $n$ in our ring is invertible for the multiplication.
Where they sloppy or is there some relation I have missed between the existence of some primitive $n$-th root of unity and the ring element $n=1+\ldots+1$?