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GRADED INTEGRAL DOMAINS AND PRUFER-LIKE DOMAINS

장규환
대한수학회 (The Korean Mathematical Society)|2017.11|대한수학회지 (Journal of the KMS)|제54권 제6호|pp. 1733-1757

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참고문헌

키워드

graded integral domain$D+XK[X]$(graded) UMT-domain(graded) Pr\"ufer domain
주제분류수학

초록

Let $R =\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be an integral domain graded by an arbitrary torsionless grading monoid $\Gamma$, $\bar{R}$ be the integral closure of $R$, $H$ be the set of nonzero homogeneous elements of $R$, $C(f)$ be the fractional ideal of $R$ generated by the homogeneous components of $f \in R_H$, and $N(H) = \{f \in R\mid C(f)_v = R\}$. Let $R_H$ be a UFD. We say that a nonzero prime ideal $Q$ of $R$ is an {\em upper to zero} in $R$ if $Q = fR_H \cap R$ for some $f \in R$ and that $R$ is a {\em graded UMT-domain} if each upper to zero in $R$ is a maximal $t$-ideal. In this paper, we study several ring-theoretic properties of graded UMT-domains. Among other things, we prove that if $R$ has a unit of nonzero degree, then $R$ is a graded UMT-domain if and only if every prime ideal of $R_{N(H)}$ is extended from a homogeneous ideal of $R$, if and only if $\bar{R}_{H \setminus Q}$ is a graded-Pr\"ufer domain for all homogeneous maximal $t$-ideals $Q$ of $R$, if and only if $\bar{R}_{N(H)}$ is a Pr\"ufer domain, if and only if $R$ is a UMT-domain.

Abstract

Let $R =\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be an integral domain graded by an arbitrary torsionless grading monoid $\Gamma$, $\bar{R}$ be the integral closure of $R$, $H$ be the set of nonzero homogeneous elements of $R$, $C(f)$ be the fractional ideal of $R$ generated by the homogeneous components of $f \in R_H$, and $N(H) = \{f \in R\mid C(f)_v = R\}$. Let $R_H$ be a UFD. We say that a nonzero prime ideal $Q$ of $R$ is an {\em upper to zero} in $R$ if $Q = fR_H \cap R$ for some $f \in R$ and that $R$ is a {\em graded UMT-domain} if each upper to zero in $R$ is a maximal $t$-ideal. In this paper, we study several ring-theoretic properties of graded UMT-domains. Among other things, we prove that if $R$ has a unit of nonzero degree, then $R$ is a graded UMT-domain if and only if every prime ideal of $R_{N(H)}$ is extended from a homogeneous ideal of $R$, if and only if $\bar{R}_{H \setminus Q}$ is a graded-Pr\"ufer domain for all homogeneous maximal $t$-ideals $Q$ of $R$, if and only if $\bar{R}_{N(H)}$ is a Pr\"ufer domain, if and only if $R$ is a UMT-domain.

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