Hwankoo Kim

w -Modules over Commutative Rings

In 1977, Glaz and Vasconcelos [73] introduced the concept of semidivisorial modules to study some properties of flat modules.

Module-Theoretic Characterizations of t-Linkative Domains

If R is an integral domain, we extend to any R-module the notion of semi-divisorial envelope, or w-envelope, defined by Wang and McCasland for torsion-free R-modules, and introduce and study the related notions of co-semi-divisoriality and w-nullity. These concepts are then used to give new module-theoretic characterizations of t-linkative domains, a class of domains widely considered in multiplicative ideal theory.

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

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.

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

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].

INJECTIVE MODULES OVER w-NOETHERIAN RINGS, II

Abstract. By utilizing known characterizations of w-Noetherian rings interms of injective modules, we give more characterizations of w-Noether-ian rings. More precisely, we show that a commutative ring R is w-Noetherian if and only if the direct limit of GV-torsion-free injective R-modules is injective; if and only if every R-module has a GV-torsion-freeinjective (pre)cover; if and only if the direct sum of injective envelopes ofw-simple R-modules is injective; if and only if the essential extension ofthe direct sum of GV-torsion-free injective R-modules is the direct sumof GV-torsion-free injective R-modules; if and only if every F w,f (R)-injective w-module is injective; if and only if every GV-torsion-free R-module admits an i-decomposition. 1. IntroductionFor the last few decades, characterizingNoetherian rings in terms ofinjectivemodules has drawn considerable attention from many algebraists. Matlis ([19]),Papp ([20]), Bass ([2]), Faith and Walker ([8]), Kurshan ([18]), Goursaud andValette ([11]), Beidar and Ke ([4]), and Beidar, Jain and Srivastava ([3]) havedone much meaningful work in this field. Since the birth of the theory of staroperations, heavy concentration has been put on ideal theory. Even so, we stillhope that the theory of star operations can play a role in researching the directsum representations of injective modules and related topics [14]. Inspired bythe study on injective modules over Noetherian rings, some researchers havepaid attention to the studies on injective modules over w-Noetherian rings.In [26], Yin et al. defined a w-Noetherian ring as a commutative ring whichsatisfies the ascending chain condition of w-ideals. As for the integral domain, aw-Noetherian ring actually is a strong Mori domain. In 2005, Fuchs provedthatthe integral domain R is a strong Mori domain if and only if E(Q/R) is a Σ-injective module [9]. According to the Cartan-Eilenberg-Bass-Papp Theorem,R is a Noetherian ring if and only if the direct sum of injective modules isinjective. In 2008, Kim et al. proved that the integral domain R is a strong

w-INJECTIVE MODULES AND w-SEMI-HEREDITARY RINGS

Abstract. Let Rbe a commutative ring with identity. An R-module Mis said to be w-projective if Ext 1R (M,N) is GV-torsion for any torsion-freew-module N. In this paper, we define a ring R to be w-semi-hereditaryif every finite type ideal of R is w-projective. To characterize w-semi-hereditary rings, we introduce the concept of w-injective modules andstudy some basic properties of w-injective modules. Using these concepts,we show that Ris w-semi-hereditary if and only if the total quotient ringT(R) of R is a von Neumann regular ring and R m is a valuation domainfor any maximal w-ideal mof R. It is also shown that a connected ring Ris w-semi-hereditary if and only if Ris a Pru¨fer v-multiplication domain. 1. IntroductionThroughout, R denotes a commutative ring with identity 1 and E(M) de-notes the injective hull (or envelope) of an R-module M. And let us regardthat the v-, t- and w-operation are well-known star-operations on domains. Forunexplained terminologies and notations, we refer to [3, 14, 15].Pru¨fer v-multiplication domains (PVMD for short) have received a good dealof attention in much literature. A domain R is called a PVMD if every nonzerofinitely generated ideal I is t-invertible, that is, there is a fractional ideal B ofR such that (IB)

New key management systems for multilevel security

In this paper, we review briefly Akl and Taylor’s cryptographic solution of multilevel security problem. We propose new key management systems for multilevel security using various one-way functions.

Module-Theoretic Characterizations of Generalized GCD Domains

We give several module-theoretic characterizations of generalized GCD domains. For example, we show that an integral domain R is a generalized GCD domain if and only if semi-divisoriality and flatness are equivalent for torsion-free R-modules if and only if every w-finite w-module is projective if and only if R is w-Prüfer (in the sense of Zafrullah). We also characterize when a pullback R of a certain type is a generalized GCD domain. As an application, we characterize when R = D + XE[X] (here, D ⊆ E is an extension of domains and X is an indeterminate) is a generalized GCD domain.

Fuzzy star-operations on an integral domain

In this paper, we introduce the concept of fuzzy star-operations on an integral domain and show that the set of all fuzzy star-operations on the integral domain forms a complete lattice. We also characterize Prufer domains, psuedo-Dedekind domains, (generalized-) greatest common divisor domains, and other integral domains in terms of the invertibility of certain fractionary fuzzy ideals.

Super Finitely Presented Modules and Gorenstein Projective Modules

Let R be a commutative ring. An R-module M is said to be super finitely presented if there is an exact sequence of R-modules where each Pi is finitely generated projective. In this article, it is shown that if R has the property (B) that every super finitely presented module has finite Gorenstein projective dimension, then every finitely generated Gorenstein projective module is super finitely presented. As an application of the notion of super finitely presented modules, we show that if R has the property (C) that every super finitely presented module has finite projective dimension, then R is K0-regular, i.e., K0(R[x1,…, xn]) ≅ K0(R) for all n ≥ 1.