On finitely stable domains, II

Abstract

Among other results, we prove the following:

1. A locally Archimedean stable domain satisfies accp.

2. 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 $R^{\prime }$ of $R$ in its quotient field (this result is related to Ohm’s theorem for Prüfer domains).

3. An Archimedean stable domain $R$ is one-dimensional if and only if $R^{\prime }$ is equidimensional (generally, an Archimedean stable local domain is not necessarily one-dimensional).

4. 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.