三角矩阵环上的Gorenstein AC-投射模

Gorenstein AC-Projective Modules over Triangular Matrix Rings

考虑三角矩阵环上的Gorenstein AC-投射模.设T=(A 0 U B)是三角矩阵环,其中A和B是环,U是(B,A)-双模.证明:若BU是平坦模,U A是有限生成投射模,则左T-模M=(M_(1) M_(2))φM[KG*8]是Gorenstein AC-投射模当且仅当M 1是Gorenstein AC-投射左A-模,φM:U AM 1→M 2是单同态,且CokerφM是Gorenstein AC-投射左B-模.

We consider Gorenstein AC-projective modules over triangular matrix rings.Let T=(A 0 U B) be a triangular matrix rings,where A and B are rings and U is a(B,A)-bimodule.We prove that if BU is a flat module and U A is a finitely generated projective module,then a left T-module (M_(1) M_(2))φM is a Gorenstein AC-projective module if and only if M 1 is a Gorenstein AC-projective left A-module,φM:U AM 1→M 2 is a monomorphism,and CokerφM is a Gorenstein AC-projective left B-module.