Injective and coherent endomorphism rings relative to some matrices

Abstract

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