Commutative rings I

Two generalizations of projective modules and their applications

Two Star Operations and Their Induced Lattices

. In this paper, we introduce and study the w-weak globaldimension w-w.gl.dim(R) of a commutative ring R. As an application, itis shown that an integral domain R is a Pru¨fer v-multiplication domainif and only if w-w.gl.dim(R) 6 1. We also show that there is a largeclass of domains in which Hilbert’s syzygy Theorem for the w-weak globaldimension does not hold. Namely, we prove that if Ris an integral domain(but not a ﬁeld) for which the polynomial ring R[x] is w-coherent, thenw-w.gl.dim(R[x]) = w-w.gl.dim(R). 1. IntroductionRecall that a Pru¨fer domain is an integral domain in which every ﬁnitelygenerated ideal is invertible. It is well-known that the concept of Pru¨fer do-mains has played a central role in the development of the classical ideal theory.An important generalization of the Pru¨fer domain notion is that of a Pru¨ferv-multiplication domain (PVMD). This notion comes from multiplicative idealtheory, and various ideal-theoretic properties of it have been considered bymany authors, see for example [1, 2, 4, 7, 9, 10, 13, 14, 15]. From the homolog-ical algebra point of view, Pru¨fer domains are exactly the integral domains ofweak global dimension at most one. The original motivation for this work is toprovide a homological algebra characterization of PVMDs. To do so, we needthe notion of a w-ﬂat module. Recently, modules of this type have receivedattention in several papers in the literature, see for example [3, 11, 21, 22].Throughout, R denotes a commutative ring with an identity element and allmodules are unitary.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 isﬁnitely generated and the natural homomorphism ϕ : R → J

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THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

Let R be a commutative ring with identity and 
$$S\subseteq R$$

 a multiplicative subset. In this paper we introduce the concept of S-prime ideal which is a generalization of prime ideal. Let P be an ideal of R disjoint with S. We say that P is an S-prime ideal of R if there exists an 
$$s\in S$$

 such that for all 
$$a,b\in R$$

 if 
$$ab\in P,$$

 then 
$$sa\in P$$

 or 
$$sb\in P$$

. We show that S-prime ideals enjoy analogs of many properties of prime ideals and we study them over S-Noetherian rings.

S -prime ideals of a commutative ring

On s -strong mori domains

Prüfer v-multiplication domains and related domains of the form D+DS[Γ∗]

Let R be a commutative ring with identity and S a multiplicative subset of R. We say that R is an S-Noetherian ring if for each ideal I of R, there exist an s ∈ S and a finitely generated ideal J of R such that sI ⊆ J ⊆ I. In this article, we study transfers of S-Noetherian property to the composite semigroup ring and the composite generalized power series ring.

S-Noetherian Properties of Composite Ring Extensions

Let D be an integral domain, D[X] be the polynomial ring over D, and w be the so-called w-operation on D. We define D to be a w-LPI domain if every nonzero w-locally principal ideal of D is w-invertible. This article presents some properties of w-LPI domains. It is shown that D is a w-LPI domain if and only if D[X] is a w-LPI domain, if and only if D[X] N v is an LPI domain, where N v  = {f ∈ D[X] | c(f)−1 = D}. As a corollary, it is also shown that D is a w-LPI domain if and only if each w-locally finitely generated and w-locally free submodule of D n of rank n over D is of w-finite type. We show that if is a finite character intersection of t-linked overrings D α and if each D α is a w-LPI domain, then D is a w-LPI domain.

Integral Domains in which Every Nonzero t-Locally Principal Ideal is t-Invertible

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ⊊ ℕ0, D[Γ] be the semigroup ring of Γ over D (and hence D ⊊ D[Γ] ⊊ D[X]), and D + X n K[X] = {a + X n g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X n K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X n K[X].

Jung Wook Lim

Papers

On s -strong mori domains

Prüfer v-multiplication domains and related domains of the form D+DS[Γ∗]

S-Noetherian Properties of Composite Ring Extensions

Integral Domains in which Every Nonzero t-Locally Principal Ideal is t-Invertible

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains