ω-MODULES OVER COMMUTATIVE RINGS

Let R be a commutative ring and let M be a GV -torsionfree R-module. Then M is said to be a <TEX>ω</TEX>-module if <TEX>ExtR¹</TEX>(R/J, M) = 0 for any J <TEX>∈</TEX> GV (R), and the w-envelope of M is defined by <TEX>Mω</TEX> = {x <TEX>∈</TEX> E(M) | Jx <TEX>⊆</TEX> M for some J <TEX>∈</TEX> GV (R)}. In this paper, <TEX>ω</TEX>-modules over commutative rings are considered, and the theory of <TEX>ω</TEX>-operations is developed for arbitrary commutative rings. As applications, we give some characterizations of <TEX>ω</TEX>-Noetherian rings and Krull rings.