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Flat dimensions of injective modules over domains

임정욱|Kui Hu|De Chuan Zhou
대한수학회 (The Korean Mathematical Society)|2020.07|대한수학회보 (Bulletin of the KMS)|제57권 제4호|pp. 1075-1081|DOI 10.4134/BKMS.b190757

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키워드

${\rm wG\mbox{-}gldim}(R)$$G$-Pr$\rm\ddot{u}$fer domain$IFD(R)$$IPD(R)$
주제분류수학

초록

Let $R$ be a domain. It is proved that $R$ is coherent when $IFD(R)\lst1$, and $R$ is Noetherian when $IPD(R)\lst1$. Consequently, $R$ is a $G$-Pr$\rm\ddot{u}$fer domain if and only if $IFD(R)\lst1$, if and only if ${\rm wG\mbox{-}gldim}(R)\lst1$; and $R$ is a $G$-Dedekind domain if and only if $IPD(R)\lst1$.

Abstract

Let $R$ be a domain. It is proved that $R$ is coherent when $IFD(R)\lst1$, and $R$ is Noetherian when $IPD(R)\lst1$. Consequently, $R$ is a $G$-Pr$\rm\ddot{u}$fer domain if and only if $IFD(R)\lst1$, if and only if ${\rm wG\mbox{-}gldim}(R)\lst1$; and $R$ is a $G$-Dedekind domain if and only if $IPD(R)\lst1$.

참고문헌 (10)

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  4. E. E. Enochs, "Copure injective resolutions, at resolvents and dimensions", Comment. Math. Univ. Carolin, 34(2), pp.203~211 (1993)
  5. Xianhui Fu, "On Copure Projective Modules and Copure Projective Dimensions", Communications in Algebra, 40(1), pp.343~359 (2012), Informa UK Limited[DOI]
  6. Henrik Holm, "Rings with finite Gorenstein injective dimension", Proceedings of the American Mathematical Society, 132(5), pp.1279~1283 (2004), American Mathematical Society (AMS)[DOI]
  7. Henrik Holm, "Gorenstein homological dimensions", Journal of Pure and Applied Algebra, 189(1-3), pp.167~193 (2004), Elsevier BV[DOI]
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  9. F. Wang, "Foundations of commutative rings and their modules" (2016), Springer
  10. Shiqi Xing, "Generalized coherent domains of self-weak injective dimension at most one", Communications in Algebra, 47(5), pp.1908~1916 (2019), Informa UK Limited[DOI]

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