# GRADED INTEGRAL DOMAINS IN WHICH EACH NONZERO HOMOGENEOUS IDEAL IS DIVISORIAL

• Chang, Gyu Whan (Department of Mathematics Education Incheon National University) ;
• Hamdi, Haleh (Department of Pure Mathematics Faculty of Mathematical Sciences University of Tabriz) ;
• Sahandi, Parviz (Department of Pure Mathematics Faculty of Mathematical Sciences University of Tabriz)
• Received : 2018.09.15
• Accepted : 2019.03.04
• Published : 2019.07.31

#### 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 $Pr{\ddot{u}}fer$ domain whose maximal ideals are invertible, if and only if R satisfies the following four conditions: (i) R is a graded-$Pr{\ddot{u}}fer$ 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.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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