模的P_n-内射维数与环的整体P_n-内射维数

The P_n-injective Dimension of Modules and the Global P_n-injective Dimension of Rings

设R是任何环,n是一固定的非负整数.模W称为P_n-内射模,是指对任何投射维数不超过n的模P,有Ext^1_R(P,W)=0(谢晋,王芳贵,熊涛.四川师范大学学报(自然科学版),2016,39(2):159-162.),引入模的P_n-内射维数和环的整体P_n-内射维数的概念,证明若l.FPD(R)<∞,则对任意n≥l.FPD(R),有l.P_ndim(R)=l.FPD(R).也引入了P_n-遗传环的概念,证明任何环都是左P1-遗传环,以及当n≥2时,R是左P_n-遗传环当且仅当l.FPD(R)≤1.

Let R be any ring,and n a fixed nonnegative integer. An R-module W is called a Pn-injective module if Ext1R（ P,W）= 0 for any R-module P with projective dimension at most n（ J. Xie,F. G. Wang,T Xiong,J Sichuan Normal University（ Natural Science）,2016,39（ 2） ： 159- 162.）. In this paper,we introduce the concepts of the Pn-injective dimension of a module and the global Pn-injective dimension of a ring. It is shown that if l. FPD（ R） ∞,then l. Pndim（ R） = l. FPD（ R） for any n≥ l. FPD（ R）. We also introduce the concept of Pn-hereditary ring,and prove that any ring is left P1-hereditary ring,when n≥ 2,R is a left Pn-hereditary ring if and only if l. FPD（ R） ≤ 1.