The lattice $R$-tors for perfect rings

Abstract

We define F in R-tors by r F σ iff the class of r-codivisible modules coincides with the class of σ -codivisible modules. We prove that if R is left perfect ring (resp. semiperfect ring) then every [r] f Є R-tors/ F (resp. [X]F and [e]F) is a complete sublattice of R-tors We describe the largest element in [r] as X(Rad R/t,(Rad R)) and the least element of [r] as e (t r(RadR)) Using these results we give a necessary and sufficient condition for the central splitting of Goldman torsion theory when R is semiperfect. We prove that for a QF ring R the least element of [X] F is the Goldie torsion theory. This can be used to prove that for a QF ring F and T are equal, where r T o iff the class of r-injective modules coincides with the class of σ-injective modules .