Jump of a domain and spectrally balanced domains
Throughout, \(R\) is a locally finite-dimensional integral domain. The authors introduce and investigate two invariants, \(j_R(a) = \inf\{\, \mathrm{ht}P - \mathrm{ht}Q \mid P\) and \(Q\) are prime ideals of \(R\) with \(Q \subset aR \subseteq P\, \}\) for \(a \in R\) a nonzero, nonunit, and \(j(R) = \sup\{\, j_R(a) \mid a \in R\text{ is a nonzero, nonunit}\,\}\). They study these two ``jump'' invariants for several classes of integral domains such as polynomial rings, power series rings, localizations, and pullbacks. They are particularly interested in the case when \(j(R) = 1\) (such domains are called spectrally balanced (SB-) domains). For example, they show that if \(R\) is a finite-dimensional Prüfer domain, then \(R[X]\) (resp., \(RX\)) is an SB-domain if and only if \(R\) is an SB-domain (resp., an SB-domain satisfying the SFT-property), and that a finite-dimensional divided domain \(R\) is a Jaffard domain if and only if \(R[X_1, \dots, X_n]\) is an SB-domain for every \(n \geq 1\).
scientific article; zbMATH DE number 6120476
scientific article