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RISS 인기검색어
- 검색키워드
- 저자 : Haleh Hamdi (검색결과 5 건)
- 무료
- 기관 내 무료
- 유료
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UPPERS TO ZERO IN POLYNOMIAL RINGS OVER GRADED DOMAINS AND UMt-DOMAINS
Hamdi, Haleh,Sahandi, Parviz Korean Mathematical Society 2018 대한수학회보 Vol.55 No.1
Let $R={\bigoplus}_{{\alpha}{\in}{\Gamma}}\;R_{\alpha}$ be a graded integral domain, H be the set of nonzero homogeneous elements of R, and ${\star}$ be a semistar operation on R. The purpose of this paper is to study the properties of $quasi-Pr{\ddot{u}}fer$ and UMt-domains of graded integral domains. For this reason we study the graded analogue of ${\star}-quasi-Pr{\ddot{u}}fer$ domains called $gr-{\star}-quasi-Pr{\ddot{u}}fer$ domains. We study several ring-theoretic properties of $gr-{\star}-quasi-Pr{\ddot{u}}fer$ domains. As an application we give new characterizations of UMt-domains. In particular it is shown that R is a $gr-t-quasi-Pr{\ddot{u}}fer$ domain if and only if R is a UMt-domain if and only if RP is a $quasi-Pr{\ddot{u}}fer$ domain for each homogeneous maximal t-ideal P of R. We also show that R is a UMt-domain if and only if H is a t-splitting set in R[X] if and only if each prime t-ideal Q in R[X] such that $Q{\cap}H ={\emptyset}$ is a maximal t-ideal.
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GRADED PRIMITIVE AND INC-EXTENSIONS
Hamdi, Haleh,Sahandi, Parviz Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.2
It is well-known that quasi-$Pr{\ddot{u}}fer$ domains are characterized as those domains D, such that every extension of D inside its quotient field is a primitive extension and that primitive extensions are characterized in terms of INC-extensions. Let $R={\bigoplus}_{{\alpha}{{\in}}{\Gamma}}$ $R_{\alpha}$ be a graded integral domain graded by an arbitrary torsionless grading monoid ${\Gamma}$ and ${\star}$ be a semistar operation on R. The main purpose of this paper is to give new characterizations of gr-${\star}$-quasi-$Pr{\ddot{u}}fer$ domains in terms of graded primitive and INC-extensions. Applications include new characterizations of UMt-domains.
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Uppers to zero in polynomial rings over graded domains and UM$t$-domains
Haleh Hamdi,Parviz Sahandi 대한수학회 2018 대한수학회보 Vol.55 No.1
Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain, $H$ be the set of nonzero homogeneous elements of $R$, and $\star$ be a semistar operation on $R$. The purpose of this paper is to study the properties of quasi-Pr\"{u}fer and UM$t$-domains of graded integral domains. For this reason we study the graded analogue of $\star$-quasi-Pr\"{u}fer domains called gr-$\star$-quasi-Pr\"{u}fer domains. We study several ring-theoretic properties of gr-$\star$-quasi-Pr\"{u}fer domains. As an application we give new characterizations of UM$t$-domains. In particular it is shown that $R$ is a gr-$t$-quasi-Pr\"{u}fer domain if and only if $R$ is a UM$t$-domain if and only if $R_P$ is a quasi-Pr\"{u}fer domain for each homogeneous maximal $t$-ideal $P$ of $R$. We also show that $R$ is a UM$t$-domain if and only if $H$ is a $t$-splitting set in $R[X]$ if and only if each prime $t$-ideal $Q$ in $R[X]$ such that $Q\cap H=\emptyset$ is a maximal $t$-ideal.
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Graded integral domains in which each nonzero homogeneous ideal is divisorial
장규환,Haleh Hamdi,Parviz Sahandi 대한수학회 2019 대한수학회보 Vol.56 No.4
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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GRADED INTEGRAL DOMAINS IN WHICH EACH NONZERO HOMOGENEOUS IDEAL IS DIVISORIAL
Chang, Gyu Whan,Hamdi, Haleh,Sahandi, Parviz Korean Mathematical Society 2019 대한수학회보 Vol.56 No.4
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.
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