Some torsion theoretical characterizations of rings (Q1066986)

Let (\({\mathcal G},{\mathcal F})\) be the Goldie torsion theory for modules over a ring R. The author looks at connections between modules over R and modules over \(\bar R=R/T(R)\), where T(R) is the torsion submodule of R. He obtains the theorems: Every torsion free R-module is quasi-injective iff \(\bar R\) is semisimple Artinian: If the maximal left quotient ring of R is semisimple Artinian and \(\bar R-\)flat, then each torsion free left R-module is quasi-projective if and only if \(\bar R\) is left perfect and left hereditary; and R is left semihereditary iff it is non-singular and each homomorphic image of a torsion free coflat module is coflat. For an arbitrary hereditary torsion theory (\({\mathcal T},{\mathcal F})\), he shows that every torsion free R-module is coflat iff \(\bar R\) is von Neumann regular.