Ω-Modules OVER COMMUTATIVE RINGS

Abstract

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

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

“…Let a be an element of R which is neither a zero-divisor nor a unit. Then aR ∼ = R. By [25,Theorem 2.7], R/aR is a GV-torsionfree R-module. Since R/aR is a torsion R-module, R/aR is not a w-flat module, and so w-fd R (R/aR) = 1.…”

“…Since c(A) w = R, there is, by [25,Proposition 3.7], a finitely generated subideal B of c(A) with B w = R. Hence there are finitely many polynomials f 1 , . .…”

“…Note that a finitely generated ideal J of R is a GV-ideal if and only if J w = R (see [25,Proposition 3.5]). From this, we have…”

“…In the recent paper [26], it was shown that all flat modules are w-modules. Let w-Max(R) denote the set of w-ideals of R maximal among proper integral w-ideals of R and we call m ∈ w-Max(R) a maximal w-ideal of R. Then by [25,Proposition 3.8] every maximal w-ideal is prime. Notice that an R-module M is GV-torsion if and only if M m = 0 for all m ∈ w-Max(R) (see [23,Theorem 2.7]).…”

“…Now, we review some definitions and notation. Let J be an ideal of R. Following [25], J is called a Glaz-Vasconcelos ideal (a GV-ideal for short) if J is finitely generated and the natural homomorphism ϕ : R → J * = Hom R (J, R) is an isomorphism. Note that the set GV(R) of GV-ideals of R is a multiplicative system of ideals of R. Let M be an R-module.…”