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.
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机译：回想一下，如果$ R $的每个素理想都是强素数，则将交换环$ R $称为伪估值环。我们定义了一个完全伪评估环。令$ R $为环（不一定是可交换的）。我们说，如果$ R $的每个主要理想都是完全质数，则$ R $是完全伪估值环。以此证明，如果$ R $是可交换的Noether环，并且也是$ mathbb {Q} $（有理数域）上的代数，而$ delta $是$ R $的导数，则$ R $是完全伪评估环，意味着$ R [x; delta]是完全伪评估环。当用最小素数替换素数时，我们证明了类似的结果。
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