On relative injective modules and relative coherent rings

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(R)(1) (P,M) = 0 for any (m,n)-presented right R-module P; M is called (m,n)-flat at if Tor(1)(R) (M,Q) = 0 for any (m,n)-presented left R-module Q. In this article, M is defined to be (m,n)-projective (resp. (m,n)-cotorsion) if Ext(R)(1) (M,N) = 0 (resp. Ext(R)(1) (N,M) = 0) for any (m,n)-injective (resp. (m,n)-flat) right R-module N. These concepts are used to characterize von Neumann regular rings and (m,n)-coherent rings. Some known results are extended.