Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian.

My question is the following problem:

> Given the dual group $\widehat{R}$ of $(R,+)$, under what conditions of $R$ is it true that

> $$
\#\{\text{primitive* characters in ($\widehat{R},\cdot$)} =\#R^\times,
$$
i.e. when does the number of primitive*
> characters of the abelian group $(R,+)$ equal the number of units in
> $R$? 


*((**Added**)). A character $\chi\in\widehat{R}$  is said to be primitive with respect to a collection $\mathcal{C}$ of subgroups of $(R, +)$ if $\chi$ does not descend to a character on any quotient group $R/S$ for all $S\in\mathcal{C}$. To descend here is canonical in the sense that $\chi$ is invariant on the elements of any coset of $R/S$.

**NOTES**\
The archetypal example is the ring $\mathbb{Z}/n\mathbb{Z}=(\{0,1,2,\dotsc,n-1\},+,\cdot)$ of integers modulo $n$ where the answer is positive when $\mathcal{C}$ contains all $\mathbb{Z}/d\mathbb{Z}$ for all $d$ divides $n$.