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