P_∞-内射模及其刻画

The Characterizations on P_∞-injective Modules

设R是任何环,模D称为P∞-内射模,是指对任何投射维数有限的模P,有Ext1R(P,D)=0.证明了(P∞,D∞)构成一个余挠理论当且仅当l.FPD(R)<∞,其中P∞表示投射维数有限的模类,D∞表示P∞-内射模类;还证明了若l.gl.dim(R)<∞,则每个P∞-内射模是内射模;最后证明了每个R-模是P∞-内射模当且仅当l.FPD(R)=0.

Let R be a ring. An R-module D is called a P∞-injective module if ExtR^1 （P, D） = 0 for any R-module P with finite pro- jective dimension. In this paper, we prove that （P∞ ,D∞） is a Cotorsion theory if and only if 1. FPD（R） 〈∞, where P∞ is the class of all R-modules with finite projective dimension, and D∞ is the class of all P∞ -injective modules. It is also shown that if 1. gl. dim（R） 〈 ∞ , then every P∞-injective module is injective. Finally, we prove that every R-module is P∞-injective module if and only if 1. FPD（R） =0.