Let $R$ be a Noetherian ring, and consider the constant $R$-group scheme $G = (\mathbf{Z}/n\mathbf{Z})_R$. I know that the Cartier dual of $G$ is the corresponding diagonalizable group scheme over $R$, and is therefore isomorphic to $\mu_{n, R}$. Is there a concrete way in terms of $n$ and $R$ to determine whether $(\mathbf{Z}/n\mathbf{Z})_R \cong \mu_{n, R}$ as group schemes? When $R$ is a field it isn't too hard to work out (e.g. What is difference between constant group scheme associated with cyclic group and $\mu_n$), but for general rings it seems a little bit more complicated, for instance because the group of $R$-points of $(\mathbf{Z}/n\mathbf{Z})_R$ is not necessarily isomorphic to $\mathbf{Z}/n\mathbf{Z}$ unless $\mathrm{Spec} R$ is connected. And in this general setting one no longer has access to the Galois theory (or at least factorization of polynomials) available over fields.
I don't think that the answer can be that it is true if and only if $R$ has a primitive $n$-th root of unity (see the comments).