Injective and coherent endomorphism rings relative to some matrices (Q6083292)

In this paper author discussed about, for given two cardinal numbers \(\alpha\) and \(\beta\) and a row-finite matrix \(A \in RFM_{\beta \times \alpha} (S)\), a right \(R\)-module \(M\) with \(S = \mathrm{End}(M_R),\) \(_{S}M\) is called injective relative to \(A\) if every left \(S\)-homomorphism from \(S^{(\beta)}A\) to \(M\) extends to one from \(S^{(\alpha)}\) to \(M\). The author proved that \(_{S}M\) is injective relative to \(A\) if and only if \(\mathrm{Ext}_S^1( S^{(\alpha)}/ S^{(\beta)}A, M) = 0\) if and only if the right R-module \(M^{(\beta)}/AM^{(\alpha)}\) is co-generated by \(M\) in Theorem 2.2 and in Lemma 2.8, shown that, if \(_{S}S\) is injective relative to \(A\) then \(M\) is quasi-projective relative to \(A\). Theorem 2.10, claim that, \(S\) is left injective relative to \(A\) if and only if \(M\) is quasi-projective relative to \(A\). Also proved that, \(S\) is left coherent relative to \(A\) if and only if \(\Hom_R (M^n/AM^{(\alpha)} , M)\) is a finitely generated left \(S\)-module if and only if \(M^n/AM^{(\alpha)}\) has an \(\mathrm{add}(M)\)-preenvelope, Theorem 3.5. He has given the necessary and sufficient conditions under which \(M^n/AM^{(\alpha)}\) has an \(\mathrm{add}(M)\)-preenvelope, which is monic (resp., epic, having the unique mapping property). Also new characterizations of left \(n\)-semihereditary rings and von Neumann regular rings are given. Finally proved that \(S^{(n)}A\) is a projective left \(S\)-module and \(_{S}M\) is injective relative to \(A\) if and only if \(M^n/AM^{(\alpha)}\) is a direct summand of \(M^n\). During reviewing, it is found that article is a fine piece of research work which are presented in a well and organised way.