杨瑞, 王淼, 王占平. 3阶三角矩阵环上的Gorenstein投射模及其维数[J]. 云南大学学报(自然科学版), 2022, 44(5): 888-894. doi: 10.7540/j.ynu.20210215
引用本文: 杨瑞, 王淼, 王占平. 3阶三角矩阵环上的Gorenstein投射模及其维数[J]. 云南大学学报(自然科学版), 2022, 44(5): 888-894. doi: 10.7540/j.ynu.20210215
YANG Rui, WANG Miao, WANG Zhan-ping. Gorenstein projective modules and dimensions over triangular matrix ring of order 3[J]. Journal of Yunnan University: Natural Sciences Edition, 2022, 44(5): 888-894. DOI: 10.7540/j.ynu.20210215
Citation: YANG Rui, WANG Miao, WANG Zhan-ping. Gorenstein projective modules and dimensions over triangular matrix ring of order 3[J]. Journal of Yunnan University: Natural Sciences Edition, 2022, 44(5): 888-894. DOI: 10.7540/j.ynu.20210215

3阶三角矩阵环上的Gorenstein投射模及其维数

Gorenstein projective modules and dimensions over triangular matrix ring of order 3

  • 摘要: T = \left( \beginarray*20c A_1&0&0 \\ U_21&A_2&0 \\ U_31&U_32&A_3 \endarray \right) 是3阶三角矩阵环,其中 A_i 是环 (i = 1,2,3) U_ij (A_i,A_j) -双模 (1 \leqslant j < i \leqslant 3) . 探讨了环 T 上的模是Gorenstein投射模的等价条件. 设 M = \left( \beginarray*20c M_1 \\ M_2 \\ M_3 \endarray \right)_\varphi ^M 是左 T -模. 若 U_ij (1 \leqslant j < i \leqslant 3) 作为左 A_i -模时投射维数有限, U_32 作为右 A_2 -模时平坦,以及 U_i1(i = 2,3) 作为右 A_1 -模时平坦维数有限,则 M 是Gorenstein投射左 T -模当且仅当 M_1 是Gorenstein投射左 A_1 -模,以及对每个i = 1,2, \varphi _i^M:U_i + 1,i \otimes _A_iM_i \to M_i + 1是单同态,且 \rmcoker \varphi _i^M 是Gorenstein投射左 A_i + 1 -模. 同时刻画了左 T -模 M 的Gorenstein 投射维数.

     

    Abstract: Let T = \left( \beginarray*20c A_1&0&0 \\ U_21&A_2&0 \\ U_31&U_32&A_3 \endarray \right) be triangular matrix ring of order 3, where A_i are rings (i = 1,2,3) and U_ij are (A_i,A_j) -bimodule (1 \leqslant j < i \leqslant 3) . We obtain equivalent condition that module over ring T is Gorenstein projective module. Suppose M = \left( \beginarray*20c M_1 \\ M_2 \\ M_3 \endarray \right)_\varphi ^M is left T -module. If U_ij(1 \leqslant j < i \leqslant 3) as left A_i -module, they have finite projective dimensions, U_32 as right A_2 -module, it is flat, and U_i1(i = 2,3) as right A_1 -module, they have finite flat dimensions, then M is Gorenstein projective left T -module if and only if M_1 is Gorenstein projective left T -module, and each i = 1,2, \varphi _i^M:U_i + 1,i \otimes _A_iM_i \to M_i + 1 is monomorphism, and \rmcoker \;\varphi _i^M is Gorenstein projective left A_i + 1 -module. At the same time, we also characterize Gorenstein projective dimensions of left T -module M.

     

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