【摘要】：本文主要研究了模的偽凝聚性和PC-內(nèi)射性確定的同調(diào)維數(shù)及其在形式三角矩陣環(huán)上的應(yīng)用.設(shè)凡是任何環(huán),若R-模N的每個有限生成子模是有限表現(xiàn)的,則稱N是偽凝聚模. R-模L稱為PC-內(nèi)射模是指對任何偽凝聚模N,有 Ext1/R(N,L) = 0.本文主要結(jié)果如下:設(shè)R是Noether環(huán),則R-模L是PC-內(nèi)射模當(dāng)且僅當(dāng)L是內(nèi)射模;設(shè)R是凝聚環(huán),則R-模L是PC-內(nèi)射模當(dāng)且僅當(dāng)對任何偽凝聚模N及任何正整數(shù)k ≥ 1,有ExtRk(N,L) = 0;本文還引入了模的PC-內(nèi)射維數(shù)和環(huán)的整體PC-內(nèi)射維數(shù)(PC-dim(R))的概念,證明了若R是凝聚環(huán),則有w.gl.dim(R)≤PC-dim(R)≤gl.dim(R)≤PC-dim(R) + 1.隨后,本文又給出凝聚環(huán)上PC-內(nèi)射維數(shù)的換環(huán)定理及其相關(guān)的維數(shù)公式,證明了若R是凝聚環(huán),則有PC-dim(R[x])=PC-dim(R)+ 1.設(shè)A,B是任何環(huán),M是A-B-雙模,則稱T=(?)是形式三角矩陣環(huán).最后,本文對形式三角矩陣環(huán)上的PC-內(nèi)射模結(jié)構(gòu)進(jìn)行刻畫,并計(jì)算了形式三角矩陣環(huán)的整體PC-內(nèi)射維數(shù),得到了若T是右凝聚環(huán),M是有限表現(xiàn)右A-模,則有Max{PC-dim(A),PC-dim(B)}≤PC-dim(T) ≤ 1 + Max{1 + PC-dim(A),PC-dim(B)};若T是 Noether 環(huán),則有]Max{gl.dim(A),gl.dim(B)}≤gl.dim(T)≤1+Max{1+gl.dim(A),gl.dim(B)};特別地,若T是Noether環(huán),M是平坦右A-模,則有Max{gl.dim(A),gl.dim(B)}≤gl.dim(T) 1 + Max{gl.dim(A), gl.dim(B)}。
[Abstract]:In this paper, we mainly study the homological dimension of the pseudo-cohesion and PC- injectivity of modules and their applications in formal triangular matrix rings. If every finitely generated submodule of R-module N is finitely represented, then N is called a pseudo-coherent module. The R-module L is called PC- injective module, which means that for any pseudo-condensed module N, there is Ext1/R (N, L) = 0. The main results are as follows: let R be a Noether ring, then R-module L is an PC- injective module if and only if L is an injective module; Let R be a coherent ring, then R-module L is a PC- injective module if and only if there is ExtRk (N, L) = 0 for any pseudo-coherent module N and any positive integer k 鈮，

本文編號：2463191