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RISS 인기검색어
- 검색키워드
- 주제어 : pseudo-valuation ring (검색결과 8 건)
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ON ϕ-PSEUDO ALMOST VALUATION RINGS
Esmaeelnezhad, Afsaneh,Sahandi, Parviz Korean Mathematical Society 2015 대한수학회보 Vol.52 No.3
The purpose of this paper is to introduce a new class of rings that is closely related to the classes of pseudo valuation rings (PVRs) and pseudo-almost valuation domains (PAVDs). A commutative ring R is said to be ${\phi}$-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map ${\phi}$ from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a ${\phi}$-ring R is said to be a ${\phi}$-pseudo-strongly prime ideal if, whenever $x,y{\in}R_{Nil(R)}$ and $(xy){\phi}(P){\subseteq}{\phi}(P)$, then there exists an integer $m{\geqslant}1$ such that either $x^m{\in}{\phi}(R)$ or $y^m{\phi}(P){\subseteq}{\phi}(P)$. If each prime ideal of R is a ${\phi}$-pseudo strongly prime ideal, then we say that R is a ${\phi}$-pseudo-almost valuation ring (${\phi}$-PAVR). Among the properties of ${\phi}$-PAVRs, we show that a quasilocal ${\phi}$-ring R with regular maximal ideal M is a ${\phi}$-PAVR if and only if V = (M : M) is a ${\phi}$-almost chained ring with maximal ideal $\sqrt{MV}$. We also investigate the overrings of a ${\phi}$-PAVR.
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ON φ-PSEUDO ALMOST VALUATION RINGS
Afsaneh Esmaeelnezhad,Parviz Sahandi 대한수학회 2015 대한수학회보 Vol.52 No.3
The purpose of this paper is to introduce a new class of rings that is closely related to the classes of pseudo valuation rings (PVRs) and pseudo-almost valuation domains (PAVDs). A commutative ring R is said to be a φ-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map φ from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a φ-ring R is said to be a φ-pseudo-strongly prime ideal if, whenever x, y ∈ RNil(R) and (xy)φ(P) ⊆ φ(P), then there exists an integer m ≥ 1 such that either xm ∈ φ(R) or ymφ(P) ⊆ φ(P). If each prime ideal of R is a φ-pseudo strongly prime ideal, then we say that R is a φ-pseudo-almost valuation ring (φ-PAVR). Among the properties of φ-PAVRs, we show that a quasilocal φ-ring R with regular maximal ideal M is a φ-PAVR if and only if V = (M : M) is a φ-almost chained ring with maximal ideal √MV . We also investigate the overrings of a φ-PAVR.
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El Khalfi, Abdelhaq,Kim, Hwankoo,Mahdou, Najib Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.4
The purpose of this paper is to introduce a new class of rings that is closely related to the class of pseudo-Krull domains. Let 𝓗 = {R | R is a commutative ring and Nil(R) is a divided prime ideal of R}. Let R ∈ 𝓗 be a ring with total quotient ring T(R) and define 𝜙 : T(R) → R<sub>Nil(R)</sub> by ${\phi}({\frac{a}{b}})={\frac{a}{b}}$ for any a ∈ R and any regular element b of R. Then 𝜙 is a ring homomorphism from T(R) into R<sub>Nil(R)</sub> and 𝜙 restricted to R is also a ring homomorphism from R into R<sub>Nil(R)</sub> given by ${\phi}(x)={\frac{x}{1}}$ for every x ∈ R. We say that R is a 𝜙-pseudo-Krull ring if 𝜙(R) = ∩ R<sub>i</sub>, where each R<sub>i</sub> is a nonnil-Noetherian 𝜙-pseudo valuation overring of 𝜙(R) and for every non-nilpotent element x ∈ R, 𝜙(x) is a unit in all but finitely many R<sub>i</sub>. We show that the theories of 𝜙-pseudo Krull rings resemble those of pseudo-Krull domains.
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Abdelhaq El Khalfi,김환구,Najib Mahdou 대한수학회 2020 대한수학회논문집 Vol.35 No.4
The purpose of this paper is to introduce a new class of rings that is closely related to the class of pseudo-Krull domains. Let $\mathcal{H} = \{R \,|\, R$ is a commutative ring and $\Nil(R)$ is a divided prime ideal of $R\}$. Let $R\in \mathcal{H}$ be a ring with total quotient ring $T(R)$ and define $\phi : T(R) \longrightarrow R_{\Nil(R)}$ by $\phi(\frac{a}{b}) = \frac{a}{b}$ for any $a \in R$ and any regular element $b$ of $R$. Then $\phi$ is a ring homomorphism from $T(R)$ into $R_{\Nil(R)}$ and $\phi$ restricted to $R$ is also a ring homomorphism from $R$ into $R_{\Nil(R)}$ given by $\phi(x) = \frac{x}{1}$ for every $x \in R$. We say that $R$ is a $\phi$-pseudo-Krull ring if $\phi(R) = \bigcap R_i$, where each $R_i$ is a nonnil-Noetherian $\phi$-pseudo valuation overring of $\phi(R)$ and for every non-nilpotent element $x \in R$, $\phi(x)$ is a unit in all but finitely many $R_i$. We show that the theories of $\phi$-pseudo Krull rings resemble those of pseudo-Krull domains.
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LOCALLY PSEUDO-VALUATION DOMAINS OF THE FORM D[X]<sub>Nv</sub>
Chang, Gyu-Whan Korean Mathematical Society 2008 대한수학회지 Vol.45 No.5
Let D be an integral domain, X an indeterminate over D, $N_v = \{f{\in}D[X]|(A_f)_v=D\}.$. Among other things, we introduce the concept of t-locally PVDs and prove that $D[X]N_v$ is a locally PVD if and only if D is a t-locally PVD and a UMT-domain, if and only if D[X] is a t-locally PVD, if and only if each overring of $D[X]N_v$ is a locally PVD.
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LOCALLY PSEUDO-VALUATION DOMAINS OF THE FORM D[X]Nv
장규환 대한수학회 2008 대한수학회지 Vol.45 No.5
Let D be an integral domain, X an indeterminate over D, Nv = {f ∈ D[X]|(Af )v = D}. Among other things, we introduce the concept of t-locally PVDs and prove that D[X]Nv is a locally PVD if and only if D is a t-locally PVD and a UMT-domain, if and only if D[X] is a t-locally PVD, if and only if each overring of D[X]Nv is a locally PVD. Let D be an integral domain, X an indeterminate over D, Nv = {f ∈ D[X]|(Af )v = D}. Among other things, we introduce the concept of t-locally PVDs and prove that D[X]Nv is a locally PVD if and only if D is a t-locally PVD and a UMT-domain, if and only if D[X] is a t-locally PVD, if and only if each overring of D[X]Nv is a locally PVD.
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Some Analogues of a Result of Vasconcelos
DOBBS, DAVID EARL,SHAPIRO, JAY ALLEN Department of Mathematics 2015 Kyungpook mathematical journal Vol.55 No.4
Let R be a commutative ring with total quotient ring K. Each monomorphic R-module endomorphism of a cyclic R-module is an isomorphism if and only if R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism if and only if R = K. We say that R has property (${\star}$) if for each nonzero element $a{\in}R$, each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (${\star}$), then each nonzero principal prime ideal of R is a maximal ideal, but the converse is false, even for integral domains of Krull dimension 2. An integral domain R has property (${\star}$) if and only if R has no R-sequence of length 2; the "if" assertion fails in general for non-domain rings R. Each treed domain has property (${\star}$), but the converse is false.
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