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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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참고문헌 (35)
참고문헌 신청
D. Costa / 1978 / The construction D + XDS[X] / J. Algebra 53 (2) : 423 ~ 439
D. D. Anderson / 1982 / Divisorial ideals and invertible ideals in a graded integral domain / J. Algebra 76 (2) : 549 ~ 569
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D. D. Anderson / 1991 / Splitting the t-class group / J. Pure Appl. Algebra 74 (1) : 17 ~ 37
D. D. Anderson / 1982 / Divisorial ideals and invertible ideals in a graded integral domain / J. Algebra 76 (2) : 549 ~ 569
D. D. Anderson / 1988 / On primary factorizations / J. Pure Appl. Algebra 54 (2-3) : 141 ~ 154
D. D. Anderson / 1990 / Weakly factorial domains and groups of divisibility / Proc. Amer. Math. Soc 109 (4) : 907 ~ 913
D. D. Anderson / 1991 / Splitting the t-class group / J. Pure Appl. Algebra 74 (1) : 17 ~ 37
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