Let R be a commutative ring and let M be a GV -torsionfree R-module. Then M is said to be a $omega$-module if $Ext_R^1$(R/J, M) = 0 for any J $in$ GV (R), and the w-envelope of M is defined by $M_{omega}$ = {x $in$ E(M) | Jx $subseteq$ M for some J $in$ GV (R)}. In this paper, $omega$-modules over commutative rings are considered, and the theory of $omega$-operations is developed for arbitrary commutative rings. As applications, we give some characterizations of $omega$-Noetherian rings and Krull rings.