Resolutions and dimensions of relative injective modules and relative flat modules

摘要

Let $m$ and $n$ be fixed positive integers and $M$ a right $R$-module. Recall that $M$ is said to be $(m,n)$-injective if Ext$^{1}(P,M)=0$ for any $(m,n)$-presented right $R$-module $P$; $M$ is said to be $(m,n)$-flat if Tor$_{1}(N ,P)=0$ for any $(m,n)$-presented left $R$-module $P$. In terms of some derived functors, relative injective or relative flat resolutions and dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and p.p.~rings are given.