Articles written in Proceedings – Mathematical Sciences

    • FGT-Injective Dimensions of 𝛱-Coherent Rings and almost Excellent Extension

      Yueming Xiang

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      We study, in this article, the FGT-injective dimensions of 𝛱-coherent rings. If 𝑅 is right 𝛱-coherent, and $\mathcal{T}\mathcal{I}(\mathrm{resp.}\mathcal{T}\mathcal{F})$ stands for the class of FGT-injective (resp.FGT-flat) 𝑅-modules $(n\geq 0)$, we show that the following are equivalent:

      (1) $FGT-Id_R(R)\leq n$;

      (2) If $0\to M\to F^0\to F^1\to\cdots$ is a right $\mathcal{T}\mathcal{F}$-resolution of left 𝑅-module 𝑀, then the sequence is exact at $F^k$ for $k\geq n-1$;

      (3) For every flat right 𝑅-module 𝐹, there is an exact sequence $0\to F\to A^0\to A^1\to\cdots\to A^n\to 0$ with each $A^i\in\mathcal{T}\mathcal{I}$;

      (4) For every injective left 𝑅-module 𝐴, there is an exact sequence $0\to F_n\to\cdots\to F_1\to F_0\to A\to 0$ with each $F_i\in\mathcal{T}\mathcal{F}$;

      (5) If $\cdots\to I_1\to I_0\to M\to 0$ is a minimal left $\mathcal{T}\mathcal{I}$-resolution of a right 𝑅-module 𝑀, then the sequence is exact at $I_k$ for $k\geq n-1$.

      Further, we characterize such homological dimension in terms of $\mathcal{T}\mathcal{I}-syzygy$ and $\mathcal{T}\mathcal{F}-cosyzygy$ of modules. Finally, we consider almost excellent extensions of rings. These extend the corresponding results in [10] as well.

    • Special properties of Hurwitz series rings


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      In this paper, we study some properties of the Hurwitz series ring $H R$ (resp. Hurwitz polynomial ring $h R$), such as the flatness or the faithful flatness of $H R/(f)$ (resp. $h R/(f)$), the strongly Hopfian property and the radical property of $H R$ (resp. $h R$). We give some sufficient and necessary conditions for $H R/(f)$ (resp. $h R/(f)$) to be flat or faithful flat. We also prove that the strongly Hopfian property transfer between$R$ and $H R$ (resp. $h R$), and some radicals of $H R$ can be determined in terms of those of $R$, in case $R$ satisfies some additional conditions.

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