\(\sigma^*\)-projective covers and \(\sigma\)-Artinian rings (Q1076775)

Let R be a ring with unit element, and let \(\sigma\) be a hereditary torsion theory of right R-modules. If \(P\to^{\alpha}Q\to C\to 0\) is an exact sequence with C \(\sigma\)-torsion, then \(\alpha\) is minimal if and only if Cok(M\(\to P\to^{\alpha}Q)\) is \(\sigma\)-torsion implies that Cok\((M\to P)\) is \(\sigma\)-torsion. A module P is \(\sigma^*\)-projective if every \(\sigma\)-dense submodule of P is \(\sigma\)-projective. If P and M are modules, then P is a \(\sigma^*\)-projective cover of M if P is \(\sigma^*\)-projective and there exists an exact sequence \(P\to^{\alpha}M\to C\to 0\) such that \(\alpha\) is minimal and C is \(\sigma\)-torsion. Some basic properties of \(\sigma^*\)-covers are given. If R has the descending chain condition on \(\sigma\)-closed right ideals, then every \(\sigma\)-cocritical module has a \(\sigma^*\)-projective cover. If R has the descending chain condition on \(\sigma\)-closed right ideals, then \(\sigma\) is a perfect torsion theory if and only if, for any injective cogenerator E associated with \(\sigma\), \(_{\Lambda}E\) is quasi-injective over its endomorphism ring \(\Lambda\).