三角矩阵环上的(n,d)-内射模

(n,d)-Injective modules over triangular matirx rings

设A,B是环,U是(B,A)-双模,n,d为非负整数,T=(AUOB)是形式三角矩阵环,首先,证明了M=(M_(1)M_(2)φ^(M))是n-表现左T-模当且仅当M1是n-表现左A-模,CokerφM是n-表现左B-模且φM:U■AM1→M2是单同态。其次,证明了当M=(M_(1)M_(2))φ^(M)是(n,d)-内射左T-模时,M1是(n,d)-内射左A-模,M2是(n,d)-内射左B-模。

Let A,B be rings,U a(B,A)-bimodule,n,d non-negative integers,T=(M_(1)M_(2)φ^(M))formal triangular matrix ring,firstly,it is shown that M=(M_(1)M_(2)φ^(M))is an n-presented left T-module if and only if M1is an n-presented left A-module,CokerφMis an n-presented left B-module andφM:U■AM1→M2is a monomorphism.Secondly,it is shown that M1is an(n,d)-injective left A-module and M2is an(n,d)-injective left B-module whenever M=(M_(1)M_(2))φ^(M)is an(n,d)-injective left T-module.