• Title/Summary/Keyword: Krull dimension

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A GORENSTEIN HOMOLOGICAL CHARACTERIZATION OF KRULL DOMAINS

  • Shiqi Xing;Xiaolei Zhang
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.3
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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.

ON n-ABSORBING IDEALS AND THE n-KRULL DIMENSION OF A COMMUTATIVE RING

  • Moghimi, Hosein Fazaeli;Naghani, Sadegh Rahimi
    • Journal of the Korean Mathematical Society
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    • v.53 no.6
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    • pp.1225-1236
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    • 2016
  • Let R be a commutative ring with $1{\neq}0$ and n a positive integer. In this article, we introduce the n-Krull dimension of R, denoted $dim_n\;R$, which is the supremum of the lengths of chains of n-absorbing ideals of R. We study the n-Krull dimension in several classes of commutative rings. For example, the n-Krull dimension of an Artinian ring is finite for every positive integer n. In particular, if R is an Artinian ring with k maximal ideals and l(R) is the length of a composition series for R, then $dim_n\;R=l(R)-k$ for some positive integer n. It is proved that a Noetherian domain R is a Dedekind domain if and only if $dim_n\;R=n$ for every positive integer n if and only if $dim_2\;R=2$. It is shown that Krull's (Generalized) Principal Ideal Theorem does not hold in general when prime ideals are replaced by n-absorbing ideals for some n > 1.

ON GRADED KRULL OVERRINGS OF A GRADED NOETHERIAN DOMAIN

  • Lee, Eun-Kyung;Park, Mi-Hee
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.1
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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 $\leq2$.

Some Analogues of a Result of Vasconcelos

  • DOBBS, DAVID EARL;SHAPIRO, JAY ALLEN
    • Kyungpook Mathematical Journal
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    • v.55 no.4
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    • pp.817-826
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    • 2015
  • 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.

Г-DEVIATION AND LOCALIZATION

  • Albu, Toma;Teply, Mark L.
    • Journal of the Korean Mathematical Society
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    • v.38 no.5
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    • pp.937-954
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    • 2001
  • This paper is a natural continuation of [2], [3], [4] and [5]. Localization techniques for modular lattices are developed. These techniques are applied to study liftings of linear order types from quotient lattices and to find Г-dense sets in certain lattices without Г-deviation in the sense of [4], where Г is a set of indecomposable linear order types.

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SOME CURIOSITIES OF THE ALGEBRA OF BOUNDED DIRICHLET SERIES

  • Mortini, Raymond;Sasane, Amol
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.1
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    • pp.153-161
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    • 2016
  • It is shown that the algebra $\mathfrak{H}^{\infty}$ of bounded Dirichlet series is not a coherent ring, and has infinite Bass stable rank. As corollaries of the latter result, it is derived that $\mathfrak{H}^{\infty}$ has infinite topological stable rank and infinite Krull dimension.

KRULL DIMENSION OF HURWITZ POLYNOMIAL RINGS OVER PRÜFER DOMAINS

  • Le, Thi Ngoc Giau;Phan, Thanh Toan
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.625-631
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    • 2018
  • Let R be a commutative ring with identity and let R[x] be the collection of polynomials with coefficients in R. There are a lot of multiplications in R[x] such that together with the usual addition, R[x] becomes a ring that contains R as a subring. These multiplications are from a class of functions ${\lambda}$ from ${\mathbb{N}}_0$ to ${\mathbb{N}}$. The trivial case when ${\lambda}(i)=1$ for all i gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when ${\lambda}(i)=i!$ for all i. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we completely determine the Krull dimension of $R_H[x]$ when R is a $Pr{\ddot{u}}fer$ domain. Let R be a $Pr{\ddot{u}}fer$ domain. We show that dim $R_H[x]={\dim}\;R+1$ if R has characteristic zero and dim $R_H[x]={\dim}\;R$ otherwise.