Abstract. Let R be a commutative ring and let M be a GV -torsionfree R-module. Then M is said to be a w-module if Ext 1 R (R/J, M ) = 0 for any J ∈ GV (R), and the w-envelope of M is defined by Mw = {x ∈ E(M ) | Jx ⊆ M for some J ∈ GV (R)}. In this paper, w-modules over commutative rings are considered, and the theory of w-operations is developed for arbitrary commutative rings. As applications, we give some characterizations of w-Noetherian rings and Krull rings. IntroductionLet R be a domain with quotient field K, and let F (R) be the set of nonzero fractional ideals of R.Recall from [17] that for a domain R and a torsionfree R-module M , the w-envelope of M is defined by is a * -operation called the w-operation.One can see that the notion of a w-ideal coincides with the notion of a semi-divisorial ideal introduced by Glaz and Vasconcelos in 1977 [5] which may have some far reaching effects on the theory of * -operations. As a * -operation, the w-operation was briefly yet effectively touched on by Hedstrom and Houston in 1980 under the name of F ∞ -operation [6]. Later, this * -operation was intensely studied by Wang and McCasland in a more general setting. In particular, Wang and McCasland showed that the w-envelope notion is a very useful tool in studying strong Mori domains [17,18]. For the definition of a * -operation, the reader may consult [4]. There is a considerable amount of research devoted to extending multiplicative ideal theory to commutative rings containing zero divisors, see for example

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In this paper, we concern the w-analogue of Matijevic’s result. We show that if R is a w-Noetherian ring and T a w-overring of R such that T ⊆ Rwg. Then T has ACC on regular w-ideals.

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Let R be an integral domain with quotient field K, and let X be an indeterminate over R. In this paper, we consider content formulae for power series in terms of * -operations for PVMDs, Krull domains and Dedekind domains, where * is the star-operation, d, w, t, or v. We prove that R is a Krull domain if and