兰州理工大学学报

兰州理工大学学报 ›› 2023, Vol. 49 ›› Issue (5): 167-172.

• 数理科学 • 上一篇    

三角矩阵环上投射余可解的Gorenstein AC-平坦模

秦军霞, 张翠萍*   

  1. 西北师范大学 数学与统计学院, 甘肃 兰州 730070
  • 收稿日期:2022-06-13 出版日期:2023-10-28 发布日期:2023-11-07
  • 通讯作者: 张翠萍(1974-),女,甘肃武威人,博士,副教授. Email:zhangcp@nwnu.edu.cn
  • 基金资助:
    国家自然科学基金(11761060)

Projectively coresolved Gorenstein AC-flat modules over triangular matrix rings

QIN Jun-xia, ZHANG Cui-ping   

  1. School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
  • Received:2022-06-13 Online:2023-10-28 Published:2023-11-07

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摘要： 引入投射余可解的 Gorenstein AC-平坦模的概念(简记为PGACF-模), 给出这类模的等价条件. 设$\boldsymbol{T}= \left(\begin{array}{ll}A & 0 \\ U & B\end{array}\right)$是三角矩阵环,其中A,B是环,U是B,A-双模.在一定条件下,证明了如果$M=\left(\begin{array}{l}M_{1} \\ M_{2}\end{array}\right)_{\varphi^{M}}$是PGACF左T-模,那么M1是PGACF左A-模,CokerφM是PGACF左B-模,且φM是单同态;若PGACF-模的类对扩张封闭,则上述结论反过来也成立.

关键词: 投射余可解的Gorenstein AC-平坦模, 三角矩阵环, absolutely clean模

Abstract: The concept of projectively coresolved Gorenstein AC-flat modules is introduced (PGACF-modules for short), and the equivalent condition of these modules is given. Let $\boldsymbol{T}= \left(\begin{array}{ll}A & 0 \\ U & B\end{array}\right)$ be a triangular matrix ring, where A and B are rings, and U is a (B,A)-bimodule. Under some conditions, it is demonstrated $M=\left(\begin{array}{l}M_{1} \\ M_{2}\end{array}\right)_{\varphi^{M}} $ is a PGACF left T-module, then M1 is a PGACF left A-module, CokerφM is a PGACF left B-module and the morphism φM is a monomorphism; if the class of PGACF-modules is closed under extensions, the opposite case holds.

Key words: projectively coresolved Gorenstein AC-flat modules, triangular matrix rings, absolutely clean modules

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