A PROPERTY OF WEAKLY KRULL DOMAINS D.D. ANDERSON AND MUHAMMAD ZAFRULLAH Abstract. We show that a weakly Krull domain D satisfies (∗): for every pair a, b ∈ D\{0} there is an n = n(a, b) ∈ N such that (a, b n ) is t-invertible. For D Noetherian, D satisfies (∗) if and only if every grade-one prime ideal of D is of height one. We also show that a modification of (∗) can be used to characterize Noetherian domains that are one-dimensional. Let D be an integral domain with quotient field K, and let X 1 (D) denote the set of height-one prime ideals of D. The domain D is said to be a weakly Krull domain if D is a locally finite intersection of its localizations at members of X 1 (D), i.e., if D = T P ∈X 1 (D) D P and this intersection is locally finite. Indeed, a weakly Krull domain D is the well-known Krull domain if we insist that D P is a discrete rank-one valuation domain for each P ∈ X 1 (D). And if we settle for a weakly Krull domain D such that D P is a valuation domain for each P ∈ X 1 (D), we get a generalized Krull domain of Ribenboim [R]. Obviously, a Noetherian domain is a weakly Krull domain if each grade-one prime ideal of D is of height one. The converse is also true. Weakly Krull domains have recently been of interest in connection with the study of factorization properties of algebraic order, see, e.g., Picavet-L’Hermitte [P]. The purpose of this short note is to bring to light the following interesting property of weakly Krull domains. Let D be a weakly Krull domain; then for every pair a, b ∈ D ∗ = D\{0}, there is a natural number n = n(a, b) such that (a, b n ) is a t-invertible ideal of D. What may be of general interest here is that (within the framework of Noetherian domains) Noetherian weakly Krull domains are actually characterized by this somewhat weak property and so are one-dimensional Noetherian domains on replacing “t-invertible” by “invertible”. To keep the note short, we shall assume a working knowledge of star-operations as provided by Gilmer’s book [G, Sections 32 and 34] and of the notion of t-invertibility as in [Z]. However, we recall for the reader’s convenience the following facts. Let F (D) denote the set of nonzero fractional ideals. Then for I ∈ F (D), I −1 = D : K I and the function I 7→ I v =(I −1 ) −1 on F (D) is a star-operation called the v- operation. The operation on F (D) defined by A 7→ A t = S {(a 1 ,a 2 ,...,a n ) v | a i ∈ I \{0}, n ∈ N} is called the t-operation. A fractional ideal I is a v-ideal (t-ideal) if I v = I (resp., I t = I ) and an ideal M maximal among integral t-ideals is a prime ideal called a maximal t-ideal. A fractional ideal I is t-invertible if (II −1 ) t = D. Finally, we add the following: (1) every minimal prime of a principal ideal is a prime t-ideal [HH], (2) if D is a weakly Krull domain, then X 1 (D) is precisely the set of maximal t-ideals of D [AMZ, Theorems 3.1 and 4.3], and (3) a finitely Date: July 31, 2002. 2000 Mathematics Subject Classification. Primary 13F05. Key words and phrases. Weakly Krull. 1