Mori–Nagata theorem

In algebra, the Mori–Nagata theorem introduced by Yoshiro Mori (1953) and Nagata (1955), states the following: let A be a noetherian reduced commutative ring with the total ring of fractions K. Then the integral closure of A in K is a direct product of r Krull domains, where r is the number of minimal prime ideals of A.

The theorem is a partial generalization of the Krull–Akizuki theorem, which concerns a one-dimensional noetherian domain. A consequence of the theorem is that if R is a Nagata ring, then every R-subalgebra of finite type is again a Nagata ring (Nishimura 1976).

The Mori–Nagata theorem follows from Matijevic's theorem.^[1]

References

1. McAdam, S. (1990), "Review: David Rees, Lectures on the asymptotic theory of ideals", Bull. Amer. Math. Soc. (N.S.), 22 (2): 315–317, doi:10.1090/s0273-0979-1990-15896-3

• Mori, Yoshiro (1953), "On the integral closure of an integral domain", Memoirs of the College of Science, University of Kyoto. Series A: Mathematics, 27: 249–256
• Nagata, Masayoshi (1955), "On the derived normal rings of Noetherian integral domains", Memoirs of the College of Science, University of Kyoto. Series A: Mathematics, 29: 293–303, MR 0097388
• Nishimura, Jun-ichi (1976). "Note on integral closures of a noetherian integral domain". J. Math. Kyoto Univ. 16 (1): 117–122.