The answer is no (in general) according to B. L. Osofsky, [Global dimension of valuation rings][1], 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 ${\rm{lD}}(R)=n$.


  [1]: http://www.ams.org/journals/tran/1967-127-01/S0002-9947-1967-0206074-0/S0002-9947-1967-0206074-0.pdf