Abstract: In this paper, we consider some generalizations of tilting torsion classes and cotilting torsion-free classes, give the definition and characterizations of n-tilting torsion classes and n-cotilting torsion-free classes, and study n-tilting preenvelopes and n-cotilting precovers. 

Key words: n-tilting torsion classes, n-cotilting torsion-free classes, preenvelopes, precovers