YUEMING XIANG
Articles written in Proceedings – Mathematical Sciences
Volume 120 Issue 2 April 2010 pp 149-161
FGT-Injective Dimensions of 𝛱-Coherent Rings and almost Excellent Extension
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.
Volume 128 Issue 4 September 2018 Article ID 0045 Research Article
Special properties of Hurwitz series rings
LUNQUN OUYANG KEXIN ZHENG QIONG ZHOU YUEMING XIANG
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.
Volume 130, 2020
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