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 condition on $R$ is it that $$\#\{primitive~characters~in~(\widehat{R},\cdot)\}=\#R^\times$$. i.e. when does the number of primitive characters of the abelian group $(R,+)$ equal the number of units in $R$.

NOTES
The archetypal example is the modulo $n$ ring $(\{0,1,2,...,n-1\},+,\cdot)$ where the answer is positive. I am not sure if the result is true for commutative rings, where possible the equality above is probably an isomorphism (of groups).