Let $R$ be a nonNoetherian ring. Is its left global dimension ${\rm{lD}}(R)$ equal to $\sup \{ {\rm{id}}(M) \mid M \text{ is a cyclic $R$module} \}$? Here $\rm{{id}}(M)$ denotes the injective dimension of $M$.

The answer is no (in general) according to B. L. Osofsky, Global dimension of valuation rings, Corollary 3. For any $1 \leq n \leq \infty$, there are examples where $$\sup \{ {\rm{id}}(M) \mid M \text{ is a cyclic $R$module} \}=1$$ and ${\rm{lD}}(R)=n$. 

