Injective and coherent endomorphism rings relative to some matrices

Abstract

Let MM be a right RR-module with S=End(MR)S={\rm{End}}\left({M}_{R}). Given two cardinal numbers α\alpha and β\beta and a row-finite matrix A∈RFMβ×α(S)A\in {{\rm{RFM}}}_{\beta \times \alpha }\left(S), SM{}_{S}M is called injective relative to AA if every left SS-homomorphism from S(β)A{S}^{\left(\beta )}A to MM extends to one from S(α){S}^{\left(\alpha )} to MM. It is shown that SM{}_{S}M is injective relative to AA if and only if the right RR-module Mβ∕AMα{M}^{\beta }/A{M}^{\alpha } is cogenerated by MM. SS is called left coherent relative to A∈Sβ×αA\in {S}^{\beta \times \alpha } if Ker(S(β)S→S(β)SA)\left({}_{S}S^{\left(\beta )}\to {}_{S}S^{\left(\beta )}A) is finitely generated. It is shown that SS is left coherent relative to AA if and only if Mn∕AMα{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 Mn∕AMα{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 nn-semihereditary rings and von Neumann regular rings are given.

Keywords

• coherent
• injective
• endomorphism ring
• preenvelope
• quasi-projective
• 16d40
• 16d50
• 16p70