Over a commutative ring $R$, a finite type locally free (in the weak sense) module for which the rank function is locally constant is projective.

If we notice that for each minimal prime $p$ of the ring, the rank function is constant on $V(p)$ the adherence of $p$ in the Zariski topology on finite locally free (weak sense) modules, then if the ring has only a finite number of minimal primes, the rank function is always locally constant on finite flat modules. Am I right ?