• 제목/요약/키워드: Krull domain

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ON GENERALIZED KRULL POWER SERIES RINGS

  • Le, Thi Ngoc Giau;Phan, Thanh Toan
    • 대한수학회보
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    • 제55권4호
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    • pp.1007-1012
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    • 2018
  • Let R be an integral domain. We prove that the power series ring R[[X]] is a Krull domain if and only if R[[X]] is a generalized Krull domain and t-dim $R{\leq}1$, which improves a well-known result of Paran and Temkin. As a consequence we show that one of the following statements holds: (1) the concepts "Krull domain" and "generalized Krull domain" are the same in power series rings, (2) there exists a non-t-SFT domain R with t-dim R > 1 such that t-dim R[[X]] = 1.

EVERY ABELIAN GROUP IS THE CLASS GROUP OF A RING OF KRULL TYPE

  • Chang, Gyu Whan
    • 대한수학회지
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    • 제58권1호
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    • pp.149-171
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    • 2021
  • Let Cl(A) denote the class group of an arbitrary integral domain A introduced by Bouvier in 1982. Then Cl(A) is the ideal class (resp., divisor class) group of A if A is a Dedekind or a Prüfer (resp., Krull) domain. Let G be an abelian group. In this paper, we show that there is a ring of Krull type D such that Cl(D) = G but D is not a Krull domain. We then use this ring to construct a Prüfer ring of Krull type E such that Cl(E) = G but E is not a Dedekind domain. This is a generalization of Claborn's result that every abelian group is the ideal class group of a Dedekind domain.

ON GRADED KRULL OVERRINGS OF A GRADED NOETHERIAN DOMAIN

  • Lee, Eun-Kyung;Park, Mi-Hee
    • 대한수학회보
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    • 제49권1호
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    • pp.205-211
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    • 2012
  • Let R be a graded Noetherian domain and A a graded Krull overring of R. We show that if h-dim $R\leq2$, then A is a graded Noetherian domain with h-dim $A\leq2$. This is a generalization of the well-know theorem that a Krull overring of a Noetherian domain with dimension $\leq2$ is also a Noetherian domain with dimension $\leq2$.

PRIME FACTORIZATION OF IDEALS IN COMMUTATIVE RINGS, WITH A FOCUS ON KRULL RINGS

  • Gyu Whan Chang;Jun Seok Oh
    • 대한수학회지
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    • 제60권2호
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    • pp.407-464
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    • 2023
  • Let R be a commutative ring with identity. The structure theorem says that R is a PIR (resp., UFR, general ZPI-ring, π-ring) if and only if R is a finite direct product of PIDs (resp., UFDs, Dedekind domains, π-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations v or t as follows: An integral domain R is a Krull domain if and only if every nonzero proper principal ideal of R can be written as a finite v- or t-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation u on R, so that R is a general Krull ring if and only if every proper principal ideal of R can be written as a finite u-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.

The schur group of a Krull domain

  • Shin, Kyung-Hee;Lee, Hei-Sook
    • 대한수학회논문집
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    • 제10권3호
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    • pp.527-539
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    • 1995
  • We consider the Schur groups of some module categories, which are subcategories of category of divisorial modules over a Krull domain. Then we obtain the exact sequence connecting class group, Schur class group and Schur groups of these categories.

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MODULE-THEORETIC CHARACTERIZATIONS OF KRULL DOMAINS

  • Kim, Hwan-Koo
    • 대한수학회보
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    • 제49권3호
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    • pp.601-608
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    • 2012
  • The following statements for an infra-Krull domain $R$ are shown to be equivalent: (1) $R$ is a Krull domain; (2) for any essentially finite $w$-module $M$ over $R$, the torsion submodule $t(M)$ of $M$ is a direct summand of $M$; (3) for any essentially finite $w$-module $M$ over $R$, $t(M){\cap}pM=pt(M)$, for all maximal $w$-ideal $p$ of $R$; (4) $R$ satisfies the $w$-radical formula; (5) the $R$-module $R{\oplus}R$ satisfies the $w$-radical formula.

BRAUER GROUP OVER A KRULL DOMAIN

  • Lee, Heisook
    • 대한수학회보
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    • 제26권2호
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    • pp.135-137
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    • 1989
  • Let R be a Krull domain with field of fractions K. By Br(R) we denote the Brauer group of R. Studying the Kernel of the homomorphism Br(R).rarw.Br(K), Orzech defined Brauer groups Br(M) for different categories M of R-modules [4]. In this paper we show that an algebra A in Br(D) is a maximal order in A K and that the map Br(D).rarw. Br(K) is one to one. We note here few conventions. All rings are Krull domains and all modules will be unitary. By Z we donote the set of height one prime ideals of a Krull domain.

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A GORENSTEIN HOMOLOGICAL CHARACTERIZATION OF KRULL DOMAINS

  • Shiqi Xing;Xiaolei Zhang
    • 대한수학회보
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    • 제61권3호
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    • pp.735-744
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    • 2024
  • In this note, we shed new light on Krull domains from the point view of Gorenstein homological algebra. By using the so-called w-operation, we show that an integral domain R is Krull if and only if for any nonzero proper w-ideal I, the Gorenstein global dimension of the w-factor ring (R/I)w is zero. Further, we obtain that an integral domain R is Dedekind if and only if for any nonzero proper ideal I, the Gorenstein global dimension of the factor ring R/I is zero.