Examples of Jaffard domains
Let A be a commutative domain of finite Krull dimension. The valuative dimension \(\dim_ vA\) is an integer n for which every overring of A in the quotient field \(K=Frac(A)\) has dimension \(\leq n\) and there exists an overring of A in K of dimension n or, equivalently, for which dim A[X\({}_ 1,...,X_ n]=2n\). This notion was introduced by \textit{P. Jaffard} [``Theorie de la dimension dans les anneaux de polynômes'', Mém. Sci. Math. 146 (1960; Zbl 0096.025)]. It has the property \(\dim_ vA[X_ 1,...,X_ n]=n+\dim_ vA\) which is not satisfied generally by the Krull dimension dim. In general, \(\dim (A)\leq \dim_ vA\) and the difference \(\dim_ vA-\dim (A)\) can be arbitrarily large. So it is quite natural to study rings A with \(\dim (A)=\dim_ vA\). Such a ring is called a Jaffard domain. This paper presents examples of Jaffard domains in order to point out the connection between Jaffard domains, Kaplansky strong S-domains, universally strong S-domains and UFD.
scientific article; zbMATH DE number 4071055