Injective and coherent endomorphism rings relative to some matrices

Let M be a right R-module with S = End(M-R). Given two cardinal numbers alpha and beta and a row-finite matrix A is an element of RFM beta x alpha(S), M-S is called injective relative to A if every left S-homomorphism from S((beta))A to M extends to one from S-(alpha) to M. It is shown that M-S is injective relative to A if and only if the right R-module M-beta/AM(alpha) is cogenerated by M. S is called left coherent relative to A is an element of S-beta x alpha if Ker(S-S((beta)) -> (S)S((beta))A) is finitely generated. It is shown that S is left coherent relative to A if and only if M-n/AM(alpha) has an add(M)-preenvelope. As applications, we obtain the necessary and sufficient conditions under which M-n/AM(alpha) has an add(M)-preenvelope, which is monic (resp., epic, having the unique mapping property). New characterizations of left n-semihereditary rings and von Neumann regular rings are given.