On finitely stable domains, I

Abstract

Among other results, we prove the following: A locally Archimedean stable domain satisfies accp. A stable domain R is Archimedean if and only if every nonunit of R belongs to a height-one prime ideal of the integral closure R ′ of R in its quotient field (this result is related to Ohm’s theorem for Prufer domains). An Archimedean stable domain R is one-dimensional if and only if R ′ is equidimensional (generally, an Archimedean stable local domain is not necessarily one-dimensional). An Archimedean finitely stable semilocal domain with stable maximal ideals is locally Archimedean, but generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.