PC-内射模及其刻画

The characterizations on PC-injective modules

设R是任何环,M是左R-模。M称为伪凝聚模,是指M的每个有限生成子模是有限表现的。设N是R-模,若对R的任意伪凝聚模M,有Ext1R(M,N)=0,则称N是PC-内射模。引入模的PC-内射维数和环的整体PC-内射维数,证明在凝聚环条件下PC-内射模的内射维数至多为1;对任何环R,若每一个模是PC-内射模,则伪凝聚模是投射模等。给出在凝聚环条件下环的弱整体维数、整体维数和PC-内射维数的关系。

Let R be a ring, and M an R-module. M is called a pseudo-coherent module if every finitely generated submodule of M is finitely presented. Let N be an R-module, N is called PC-injective if ExtR1 （M,N） = 0 for every pseudo-coherent module M. PC-injective dimensions of modules and the global PC- injective dimensions of tings are introduced. In a pseudo-coherent ring, it is proved that if N is a PC-in- jective, then/dRN≤ 1. For any commutative ring R, if every module is PC-injective, then pseudo-coher- ent is projective module. From these results, the connections of dimensions among gl. dim （R）, w. gl. dim（R） and PC-dim（R） are given.