Graded integral domains in which each nonzero homogeneous ideal is divisorial

Bull. Korean Math. Soc. Published online 2019 May 16

Gyu Whan Chang, Haleh Hamdi, and Parviz Sahandi
Incheon National University, University of Tabriz

Abstract : Let $\Gamma$ be a nonzero commutative cancellative monoid (written additively),
$R = \bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a $\Gamma$-graded integral domain
with $R_{\alpha} \neq \{0\}$ for all $\alpha \in \Gamma$, and $S(H) = \{f \in R \mid C(f) = R\}$.
In this paper, we study homogeneously divisorial domains which are
graded integral domains whose nonzero homogeneous ideals are divisorial.
Among other things, we show that if $R$ is integrally closed, then $R$ is a homogeneously divisorial domain
if and only if $R_{S(H)}$ is an h-local Pr\"ufer domain whose maximal ideals are invertible,
if and only if $R$ satisfies the following four conditions: (i) $R$ is a graded-Pr\"{u}fer domain,
(ii) every homogeneous maximal ideal of $R$ is invertible, (iii)
each nonzero homogeneous prime ideal of $R$ is contained in a unique homogeneous maximal ideal, and
(iv) each homogeneous ideal of $R$ has only finitely many minimal prime ideals.
We also show that if $R$ is a graded-Noetherian domain, then $R$ is a homogeneously
divisorial domain if and only if $R_{S(H)}$ is a divisorial domain of (Krull) dimension one.