Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: $(m,n)$-coherent ring, $(m,n)$-injective module, $(m,n)$-flat module, (pre)cover, (pre)envelope

MSC numbers: 16D50, 16E50, 16E10