Graded integral domains in which each nonzero homogeneous ideal is divisorial
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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 $S(H) = \{f \in R \,|\, C(f) = R\}$. 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.
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 $S(H) = \{f \in R \,|\, C(f) = R\}$. 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.
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