On locally AGCD domains

Abstract

Let [Formula: see text] be an integral domain with quotient field [Formula: see text]. Call [Formula: see text] an almost GCD domain (AGCD domain) if for all [Formula: see text], there is an integer [Formula: see text] such that [Formula: see text] is principal. We say that [Formula: see text] is a locally AGCD domain (resp., strongly locally AGCD domain) if [Formula: see text] is an AGCD domain for all maximal ideals [Formula: see text] of [Formula: see text] (resp., for all [Formula: see text], there is an integer [Formula: see text] such that [Formula: see text] is locally principal). In this paper, we study some ring-theoretic properties of locally and strongly locally AGCD domains. Let [Formula: see text] be an indeterminate over [Formula: see text]. We use the ring [Formula: see text] for an integer [Formula: see text] to give some examples of locally and strongly locally AGCD domains.