Endocoherent modules.
Recall that a left module \(M\) over an associative ring \(S\) with identity is coherent if it is finitely presented and every finitely generated submodule of \(M\) is finitely presented. Moreover, the module \(M\) is called \(\pi\)-coherent if it is finitely presented and every finitely generated left \(S\)-module which is cogenerated by \(M\) is finitely presented. Finally, for a right \(R\)-module \(M_R\) the symbol \(\text{Add\,}M\) (\(\text{add\,}M\)) denotes the category of all modules isomorphic to direct summands of (finite) direct sums of copies of \(M\). Let \(M_R\) be a right module over a ring \(R\) with the endomorphism ring \(S\). The following conditions are equivalent: (1) Every finitely generated left \(S\)-module which is cogenerated by \(_SM\) is finitely presented; (2) every finitely \(M\)-generated right \(R\)-module has an \(\text{add\,}M\)-preenvelope; (3) for every natural integer \(n\) and every subset \(X\subseteq M^n\) the annihilator \(\text{ann}_{S^{n\times n}}(X)\) of \(X\) in the matrix ring \(S^{n\times n}\) is a finitely generated left ideal (Theorem~1). If \(_SM\) is \(\pi\)-coherent, then every finitely generated module has an \(\text{add\,}M\)-preenvelope. The converse holds if \(M_R\) is finitely generated. If \(_SM\) is coherent, then every finitely presented module has an \(\text{add\,}M\)-preenvelope. The converse holds if \(M_R\) is finitely presented (Theorem~2).
scientific article; zbMATH DE number 2101118