On Pseudo-Affine Domains

概要

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In what follows, all rings considered are commutative with identity. We say that a ring A is a Hilbert ring if each prime ideal of A is an intersection of maximal ideals of R. It is known that a k-affine domain over a field k is a Hilbert ring ([G, (31.11)]). We say that a ring A is a catenary ring if the following condition is satisfied : for any prime ideals p and q of A with p⊆q, then exists a saturated chain of prime ideals starting from p and ending at q, and all such chains have the same (finite) length. We say that a ring A is a universally catenary ring if A is Noetherian and every finitely generated A-algebra is catenary. Let k be a field and R a K-affine domain. Then R is Noetherian, Hilbert and catenary. Moreover dim R_m=Tr.deg_kR<+∞ for each maximal ideal m of R. Our objective in this paper is to investigate integral domains having these properties. Throughout this paper, k denotes a field and R an integral domain containing k and K(R) denotes the quotient field of R unless otherwise specified. Any unexplained terminology is standard, as in [M], [N].
岡山理科大学の論文