[This question] is posted also on SE.

Let $R$ by a commutative ring with $1$, and $I \subset R$ a non-zero integral ideal in $R$. When $R$ has finite quotients, and $I = P$ is prime in $R$, the group of units $(R/P)^{\times}$ of the finite ring $R/P$ is cyclic as $R/P$ is a finite field. Do there exist known sufficient and necessary conditions on $R$ and $I$ in general or for certain classes of unital rings for cyclicity of $(R/I)^{\times}$ ? In particular, do there exist more general analogues of the primitive root theorem, which answers this question for $R = \mathbb{Z}$ in terms of number-theoretic criteria on the positive generators of the principal ideals $I = (n)$?

[this question]:https://math.stackexchange.com/questions/3594780/conditions-for-cyclicity-of-group-of-units-of-quotient-ring-general-analogue-of