The exponent subgroup $n\Bbb{Z}$ inducing a nontrivial $X^n - 1 = 0$ solution subgroup of $R^{\times}$ for a ring $R$ with $1$.

Question

Let $R$ be a commutative ring and $R^{\times}$ its multiplicative group.

Let $H \leqslant R^{\times}$ and $H' \subset \Bbb{Z}$ be the set of exponents $k$ such that the solutions to $X^k = 1$ form the subgroup $H$ of $R^{\times}$. $H'$ is a subgroup of $\Bbb{Z}$: $a^{k} = a^{\ell} = 1$ for all $a \in H \implies a^{k-\ell} = 1 $ for all $a \in H$.

I am interested a sufficient condition for there to exist a nontrivial $H$ induced by some $n \in \Bbb{Z}$. Would this have to do with divisors of the group or ring orders involved?