凝聚环的几个特征

Some Characteristics of Coherent Rings

证明了对于一个环R,下列条件等价:(1)R是左凝聚的;(2)对任意正整数n,Mn(R)是左1-凝聚的;(3)Ext2R(R I,N)=0对于任意有限生成左理想I及F-内射模RN成立;(4)若N1≤N都是F-内射左R-模,则N N1也是F-内射模.

For a ring R, the following statements are equivalent:(1)R is left coherent;(2)For every positive integer n, M_n(R) is left 1-coherent;(3)Ext^2_R(R/I,N)=0 holds for every finitely generated left ideal I and every F-injective module R^N;(4)If N_1≤N are both F-injective left R-modules, then N/N_1 is also F-injective.