Almost Gorenstein Dedekind domains

Abstract

An integral domain [Formula: see text] is said to be locally G-Dedekind if the Gorenstein global dimension of [Formula: see text] is at most one for each maximal ideal [Formula: see text]. In this paper, we show that an integral domain [Formula: see text] is not necessarily a G-Prüfer even if [Formula: see text] is a locally G-Dedekind domain, which gives a negative answer to the question raised by the first author. It follows that the localization of the G-Prüfer domain differs from the classical case of the Prüfer domain. We also study coherent locally G-Dedekind domains, called almost G-Dedekind domains. The almost G-Dedekind domains need not be integrally closed and fill the gap between the G-Dedekind domains and the G-Prüfer domains. Various examples are provided to illustrate the new concept.