Torsion theory extensions and finite normalizing extensions

Let S be a ring extension of R, and let and be torsion theories for right R-modules and right S-modules respectively. Using functorsHomR(−,S) and − RS, we get the connection between τR and τS, introduce torsion theory extensions related to ring extensions and establish some relations between finite normalizing extensions and torsion theory extensions. Furthermore, we consider some applications of torsion theory extensions and generalize some results of excellent extensions (almost excellent extensions) to the setting of torsion theories.