Torsion theories and primary decomposition of modules (Q1078298)

The author studies rings \(R\) which have primary decompositions in the sense that every semiartinian left \(R\)-module is a direct sum of its primary submodules. If \(R\) is left semiartinian and every left \(R\)-module has a maximal submodule, then \(R\) is shown to have primary decomposition if and only if \(\Ext(S_1,S_2)=0\) for every pair of non-isomorphic simple modules, \(S_1\), \(S_2\). In this case it is also proved that all the torsion theories of \(R\)-mod are generated by simple modules.