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 for any (m,n)-presented right R-module P; M is called (m,n)-flat if 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 (resp. ) 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.