If $R$ is a Noetherian commutative ring with unity having finitely many height one prime ideals, one could derive from the "Principal Ideal Theorem", due to Krull, that $R$ has finitely many prime ideals (all of height less than or equal to $1$). It may comes to mind that, in general,
if $R$ is a commutative Noetherian ring with unity of finite Krull dimension $d$ has finitely many prime ideals of height $n$ with $0<n≤d$, then $n=d$.
My question: is this generalization valid?
I shall appreciate any help.