When are graded integral domains almost gcd-domains

Let R = _Γ R be a (Γ-)graded integral domain and let H be the multiplicatively closed set of nonzero homogeneous elements of R. In this paper, we introduce the concepts of graded almost GCD-domains (graded AGCD-domain) and graded almost Prüfer v-multiplication domains (graded APvMD). Among other things, we show that if R is integrally closed, then (1) H is an almost lcm splitting set of R if and only if R is a graded AGCD-domain and (2) R is a graded APvMD if and only if R is a PvMD. We also give an example of a (non-integrally closed) graded AGCD-domain (respectively, graded APvMD) that is not an almost GCD-domain (respectively, almost Prüfer v-multiplication domain.