Study Of PC-Injectivity And Related Homological Properties

In this paper, we mainly study pseudo-coherent modules, PC-injective modules and their applications on formal triangular matrix rings. Let R be a ring. An R-module N is said to be pseudo-coherent if every finitely generated submodule of N is finitely presented. An R-module L is said to be PC-injective if Ext1/R(N, L) = 0 for any pseudo-coherent module N.The results of this context are the following. Let R be noetherian. An R-module L is PC-injective if and only if L is injective; if R is coherent, then R-module L is PC-injective if and only if ExtRk(N, L) = 0 for any pseudo-coherent module N and any integer k? 1. In the paper, we introduce the concepts of the dimensions of PC-injective modules and the global dimensions of PC-injectivity of rings (PC-dim(R)). We show that for a coherent ring R, the inequations w.gl.dim(R) ? PC-dim(R) ?gl.dim(R) ? PC-dim((R) + 1 hold. Also, we show the exchange theories of coherent rings and their related dimension formulae, if R is coherent, we have PC-dim(R[x]) = PC-dim(R) + 1. Let A and B be rings,and M be an A-bimodule. Then T=(?) is called formal triangularmartix ring. In the second part of the context, we mainly investigate the PC-injective modules on formal triangular matrix rings and calculate the dimensions of PC-injective modles over such rings. Let T be right coherent and let M be a finitely presented right A-module. We have Max{PC-dim(A), PC-dim(B)} ?PC-dim(T)? 1 + Max{1+PC-dim(A), PC-dim(B)}. Let T be noetheri-an. We show that Max{gl.dim(A),gl.dim(B)} ? gl.dim(T) ? 1 + Max{1 ?gl.dim(A), gl.dim(B)} ; In particular, if T is a noetherian ring and M is a flat right A-module, we prove Max{gl.dim(A),gl.dim(B)}? gl.dim(T) ?1 +Max{gl.dim(A), gl.dim(B)}.