Envelopes and covers by \(n\)-absolutely pure modules. (Q2796972)

A left \(R\)-module \(M\) is \textit{\(n\)-absolutely pure} if \(\text{Ext}_R^1(N,M)=0\) for all finitely presented left \(R\)-modules \(N\) of projective dimension \(\leq n\). Let \(\mathcal{FI}_n\) denote a class of \(n\)-absolutely pure left \(R\)-modules for an integer \(n>0\) or \(n=\infty\). Define NEWLINE\[NEWLINE\mathcal{FI}_n^\bot:=\{M\in R\text{-Mod}:\text{Ext}_R^1(A,M)=0\text{ for all }A\in\mathcal{FI}_n\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE^\bot\mathcal{FI}_n:=\{N\in R\text{-Mod}:\text{Ext}_R^1(N,B)=0\text{ for all }B\in\mathcal{FI}_n\}.NEWLINE\]NEWLINE The author shows that: 1) \((^\bot\mathcal{FI}_n,\mathcal{FI}_n)\) is a complete cotorsion theory and \(\mathcal{FI}_n\) is special preenveloping; 2) if \(R\) is a left \(n\)-coherent ring, then the cotorsion theory \((^\bot\mathcal{FI}_n,\mathcal{FI}_n)\) is hereditary and every \(R\)-module has a special \(\mathcal{FI}_n^\bot\)-envelope as well as an \(\mathcal{FI}_n\)-cover.