On \(S\)-torsion theories in \(R\)-Mod (Q5954042)

A torsion theory \((\mathcal{T,F})\) for the category \(R\)-Mod of left \(R\)-modules is called hereditary if the torsion class \(\mathcal T\) is closed under submodules. In the paper, the notion of an \(S\)-torsion theory is introduced: this is a hereditary torsion theory \((\mathcal{T,F})\) for which there exists a left ideal \(H\) of \(R\) satisfying the condition that any left ideal \(I\) of \(R\) is \(\mathcal T\)-dense in \(R\) if and only if \(H+I=R\). A ring \(R\) is said to be left semiartinian if \(\text{Soc}(M)\not=0\) for any nonzero left \(R\)-module \(M\). It is shown that if every semisimple torsion theory is an \(S\)-torsion theory then the ring \(R\) is semiartinian and right perfect. Some equivalent characterizations of semiprimitive rings in terms of \(S\)-torsion theories are pointed out.