Let R be a commutative ring with identity. If an ideal I of R can be generated by two elements then we say that / is two-generated, and if every ideal of R is twogenerated we say that R has the two-generator property. It is well known that Dedekind domains have the two-generator property and if R has the two-generator property then R has Krull dimension at most 1. In [1] Bass began the study of onedimensional rings which have the two-generator property, and showed that if such a ring R is reduced and has finite integral closure then finitely generated torsion-free Rmodules are isomorphic to direct sums of ideals. One-dimensional reduced rings having the two-generator property and finite integral closure are now often called Bass rings. According to [1, p. 19] one of the original motivations for studying the relationship of the two-generator property to decompositions of /^-modules was to understand the modules over the integral group rings Z[G], where Z is the rational integers, and G a finite abelian group, and it was pointed out there [1, p. 20] that an integral abelian group ring Z[G] has the two-generator property if and only if G has square-free order. In this paper we continue this investigation by determining the onedimensional monoid rings R[S] which have the two-generator property, where S is a commutative cancellative monoid.All rings will be assumed to be commutative, and all monoids S will be commutative and cancellative. By a DVR we mean a discrete rank one valuation domain. The monoids will be written additively, and the elements of R[S] will be written as in [3], that is, a typical element of R[S] will be denoted by r l X** +... + r n X sn , where r t eR and s t eS. We often use the augmentation map R[S] -> R. Recall that this is the i^-algebra homomorphism which maps S to the identity of R. Its kernel is called the augmentation ideal of R[S] and is generated by {X* 1 -X 9 1 g, h e S}, or by {1 -X 9 \ g e S} if S is a group. We refer to [3] for additional properties of monoid rings. The quotient group of a monoid S will be denoted G(S), and if p is a prime integer then the /?-Sylow subgroup of a finitely generated group G will be denoted G p .From the restriction on the dimension we getwhere a denotes the torsion-free rank of G(S) and dim (R) is the Krull dimension of R. Thus there are two cases to consider: dim(i?) = 1, a = 0; and dim(/?) = 0, a = 1.

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We show that a Noetherian ring R is locally quasi-unmixed if and only if for every prime ideal P e A*(ί), ht(P) = l(IR P ). The analytic spread of an e.p.f., /(/) is also defined and many of the known results for the integral closures of powers of an ideal are proven for the weak integral closures of the ideals in a strong e.p.f. Several characterizations are given of when a Noetherian ring R is locally quasi-unmixed in terms of analytic spreads and integral closure of ideals. Several applications of these equivalences are given by showing when certain prime ideals are in

Let ƒ = {In}n ≽ 0 be a filtration on a ring R, let(In)w = {x ε R; x satisfies an equation xk + i1xk − 1 + … + ik = 0, where ij ε Inj} be the weak integral closure of In and let ƒw = {(In)w}n ≽ 0. Then it is shown that ƒ ↦ ƒw is a closure operation on the set of all filtrations ƒ of R, and if R is Noetherian, then ƒw is a semi-prime operation that satisfies the cancellation law: if ƒh ≤ (gh)w and Rad (ƒ) ⊆ Rad (h), then ƒw ≤ gw. These results are then used to show that if R and ƒ are Noetherian, then the sets Ass (R/(In)w) are equal for all large n. Then these results are abstracted, and it is shown that if I ↦ Ix is a closure (resp.. semi-prime, prime) operation on the set of ideals I of R, then ƒ ↦ ƒx = {(In)x}n ≤ 0 is a closure (resp., semi-prime, prime) operation on the set of filtrations ƒ of R. In particular, if Δ is a multiplicatively closed set of finitely generated non-zero ideals of R and (In)Δ = ∪KεΔ(In, K: K), then ƒ ↦ ƒΔ is a semi-prime operation that satisfies a cancellation law, and if R and ƒ are Noetherian, then the sets Ass (R/(In)Δ) are quite well behaved.

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