A Nagata (pseudo-geometric) ring is a Noetherian ring $R$, such that $R/\mathfrak{p}$ is Japanese (N-2) for every prime $\mathfrak{p} \subseteq R$.
Matsumura claims in his book "Commutative Algebra" that
If $R$ is a Nagata ring then any localization or $R$ and any finite $R$-algebra are again Nagata.
Similarly, Nagata writes in his book "Local Rings":
If $R$ is a Nagata ring, then every homomorphic image of $R$, every ring of quotients of $R$, and every ring which is a finite module over $R$ are Nagata rings.
Both of these claims are not proven in the books and I would like to see a proof, at least of Matsumura's statement. So I would really appreciate it, if somebody here wrote it down. Thank you in advance!