JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (11): 79-86.doi: 10.6040/j.issn.1671-9352.0.2023.479

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Gorenstein flat modules and dimensions with respect to duality pairs over formal triangular matrix rings

LIU Ling, CHEN Wenjing*   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2025-11-11

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Abstract: Let T=(A 0U B) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. G F(F(R),Y ) denotes the class of all Gorenstein flat left R-modules with respect to a complete duality pair(X,Y ). Assume that(C1,C2)and(D1,D2)are complete duality pairs over the ring A and the ring B respectively, and(UC1D1,UC2,D2)is a complete duality pair over the ring T induced by(C1,C2)and(D1,D2). It is proven that, if UA has finite flat dimension and G F(F(T),UC2,D2) is closed under extensions, then a left T-module (M1M2)φM∈G F(F(T),UC2,D2) if and only if M1∈G F(F(A),C2), M2/Im(φM)∈G F(F(B),D2), and φM:U⊗AM1→M2 is a monomorphism. Furthermore, an estimate of Gorenstein flat dimension with respect to the complete duality pair(UC1D1,UC2,D2)of a left T-module is given.

Key words: formal triangular matrix ring, Gorenstein flat module with respect to a complete duality pair, Gorenstein flat dimension with respect to a complete duality pair

CLC Number: 

  • O153.3
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