Recall that a commutative ring $R$ is said to be a pseudo-valuation ring if every prime ideal of $R$ is strongly prime. We define a completely pseudo-valuation ring. Let $R$ be a ring (not necessarily commutative). We say that $R$ is a completely pseudo-valuation ring if every prime ideal of $R$ is completely prime. With this we prove that if $R$ is a commutative Noetherian ring, which is also an algebra over $\mathbb{Q}$ (the field of rational numbers) and $\delta$ a derivation of $R$, then $R$ is a completely pseudo-valuation ring implies that $R[x;\delta]$ is a completely pseudo-valuation ring. We prove a similar result when prime is replaced by minimal prime.