• Title/Summary/Keyword: regular global dimension

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THE HOMOLOGICAL PROPERTIES OF REGULAR INJECTIVE MODULES

  • Wei Qi;Xiaolei Zhang
    • Communications of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.59-69
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    • 2024
  • Let R be a commutative ring. An R-module E is said to be regular injective provided that Ext1R(R/I, E) = 0 for any regular ideal I of R. We first show that the class of regular injective modules have the hereditary property, and then introduce and study the regular injective dimension of modules and regular global dimension of rings. Finally, we give some homological characterizations of total rings of quotients and Dedekind rings.

REGULARITY RELATIVE TO A HEREDITARY TORSION THEORY FOR MODULES OVER A COMMUTATIVE RING

  • Qiao, Lei;Zuo, Kai
    • Journal of the Korean Mathematical Society
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    • v.59 no.4
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    • pp.821-841
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    • 2022
  • In this paper, we introduce and study regular rings relative to the hereditary torsion theory w (a special case of a well-centered torsion theory over a commutative ring), called w-regular rings. We focus mainly on the w-regularity for w-coherent rings and w-Noetherian rings. In particular, it is shown that the w-coherent w-regular domains are exactly the Prüfer v-multiplication domains and that an integral domain is w-Noetherian and w-regular if and only if it is a Krull domain. We also prove the w-analogue of the global version of the Serre-Auslander-Buchsbaum Theorem. Among other things, we show that every w-Noetherian w-regular ring is the direct sum of a finite number of Krull domains. Finally, we obtain that the global weak w-projective dimension of a w-Noetherian ring is 0, 1, or ∞.

ON SEMI-REGULAR INJECTIVE MODULES AND STRONG DEDEKIND RINGS

  • Renchun Qu
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.4
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    • pp.1071-1083
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    • 2023
  • The main motivation of this paper is to introduce and study the notions of strong Dedekind rings and semi-regular injective modules. Specifically, a ring R is called strong Dedekind if every semi-regular ideal is Q0-invertible, and an R-module E is called a semi-regular injective module provided Ext1R(T, E) = 0 for every 𝓠-torsion module T. In this paper, we first characterize rings over which all semi-regular injective modules are injective, and then study the semi-regular injective envelopes of R-modules. Moreover, we introduce and study the semi-regular global dimensions sr-gl.dim(R) of commutative rings R. Finally, we obtain that a ring R is a DQ-ring if and only if sr-gl.dim(R) = 0, and a ring R is a strong Dedekind ring if and only if sr-gl.dim(R) ≤ 1, if and only if any semi-regular ideal is projective. Besides, we show that the semi-regular dimensions of strong Dedekind rings are at most one.

A HOMOLOGICAL CHARACTERIZATION OF PRÜFER v-MULTIPLICATION RINGS

  • Zhang, Xiaolei
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.1
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    • pp.213-226
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    • 2022
  • Let R be a ring and M an R-module. Then M is said to be regular w-flat provided that the natural homomorphism I ⊗R M → R ⊗R M is a w-monomorphism for any regular ideal I. We distinguish regular w-flat modules from regular flat modules and w-flat modules by idealization constructions. Then we give some characterizations of total quotient rings and Prüfer v-multiplication rings (PvMRs for short) utilizing the homological properties of regular w-flat modules.

On Depth Formula and Tor Game (깊이의 식과 토르 게임에 대하여)

  • Choi Sangki
    • Journal for History of Mathematics
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    • v.17 no.4
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    • pp.37-44
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    • 2004
  • Homological algebra has emerged and developed since 1950s. However, in 1890's Hilbert investigated the resolutions in his Syzygy Theorem which is a vital ingredient in homological algebra. In 1956 Serre has proved the finite global dimension of regular local rings. His result give a basic tool in homological algebra. This paper also deals with the depth formula that was raised by Auslander in 1961.

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