GCN 环

GCN rings

首先讨论GCN环的一些性质,其次证明了如下结果:1)设R为一个环,如果R上的二阶上三角矩阵环为GCN环,则R为约化环;2)GCN的exchange环R有稳定秩1;3)R为交换环当且仅当T={a 0 b c0 a 0 da,b,c,d,0 0 a e0 0 0ae∈R}是强GCN环;4)GCN的exchange环是左quasi-duo环.

In this paper, some properties of GCN rings are discussed. Moreover, the following results are also shown： Let R be a ring. Then： 1） if the 2× 2 upper triangular matrix ring over R is a GCN ring, then R is reduced; 2） GCN exchange rings have stable range 1; 3） R is a commutativering if and only if T={（a 0 b c 0 a 0 d 0 0 a e 0 0 0 a）｜a,b,c,d,e∈R} is a GCN ring; 4） GCN exchange rings areleft quasi-duo.