Jump of a domain and spectrally balanced domains

Abstract

Let R be a locally finite-dimensional (LFD) integral domain. We investigate two invariants \({j_R(a)={{\rm inf}}\{{\rm height}P-{\rm height} Q\}}\), where P and Q range over prime ideals of R such that \({Q\subset aR\subseteq P}\), and \({j(R)={\rm sup}\{j_R(a)\}}\) (called the jump of R), where a range over nonzero nonunit elements of R. We study the jump of polynomial ring and power series ring, we give many results involving jump, and specially we give more interest to LFD-domain R such that j(R) = 1. We prove that if R is a finite-dimensional divided domain, then R is a Jaffard domain if and only if for all integer \({n,\,j(R[x_1,\ldots,x_n])=1}\).