Fanggui Wang

INDECOMPOSABLE, PROJECTIVE, AND FLATS-POSETS

Abstract For a monoid S , a (left) S -act is a nonempty set B together with a mapping S ×B→B sending (s, b) to sb such that S (tb) = lpar;st)b and 1b  = b for all S , t ∈ S and B  ∈ B. Right S -acts A can also be defined, and a tensor product A  ⊗  s B (a set)can be defined that has the customary universal property with respect to balanced maps from A × B into arbitrary sets. Over the past three decades, an extensive theory of flatness properties has been developed (involving free and projective acts, and flat acts of various sorts, defined in terms of when the tensor product functor has certain preservation properties). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by M. Kilp et al. (New York: Walter de Gruyter, 2000). To date, there have been only a few attempts to generalize this material to ordered monoids acting on partially ordered sets ( S -posets). The present paper is devoted to such a generalization. A unique decomposition theorem for S -posets is given, based on strongly convex, indecomposable S -subposets, and a structure theorem for projective S -posets is given. A criterion for when two elements of the tensor product of S -posets given, which is then applied to investigate several flatness properties.

w -Modules over Commutative Rings

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

Multiplication Modules in Which Every Prime Submodule is Contained in a Unique Maximal Submodule

Abstract Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M) = {PM | P ∈ Spec(R) and P ⊇ M ⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal.

Spectrum of a Noncommutative Ring

R is any ring with identity. Let Spec r (R) (resp. Spec(R)) be the set of all prime right ideals (resp. all prime ideals) of R and let U r (eR) = {P ∈ Spec r (R) | e ∉ P}. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec r (R) (with weak Zariski topology). A ring R is called Abelian if all idempotents in R are central (see Goodearl, 1991). A ring R is called 2-primal if every nilpotent element is in the prime radical of R (see Lam, 2001). It will be shown that for an Abelian ring R there is a bijection between the set of all idempotents in R and the clopen (i.e., closed and open) sets in Spec r (R). And the following results are obtained for any ring R: (1) For any clopen set U in Spec r (R), there is an idempotent e in R such that U = U r (eR). (2) If R is an Abelian ring or a 2-primal ring, then, for any idempotent e in R, U r (eR) is a clopen set in Spec r (R). (3) Spec r (R) is connected if and only if Spec(R) is connected.

Weak Injective and Weak Flat Modules

Let R be a ring. A left R-module M (resp., right R-module N) is called weak injective (resp., weak flat) if (resp., ) for every super finitely presented left R-module F. By replacing finitely presented modules by super finitely presented modules, we may generalize many results of a homological nature from coherent rings to arbitrary rings. Some examples are given to show that weak injective (resp., weak flat) modules need not be FP-injective (resp., not flat) in general. In addition, we introduce and study the super finitely presented dimension (denote by l.sp.gldim(R)) of R that are defined in terms of only super finitely presented left R-modules. Some known results are extended.

ON LCM-STABLE MODULES

A torsion-free module M over a commutative integral domain R is said to be LCM-stable over R if (Ra ∩ Rb)M = Ma ∩ Mb for all a, b ∈ R. We show that if the module M is LCM-stable over a GCD-domain R, then the polynomial module M[X] is LCM-stable over R[X]; if R is a w-coherent locally GCD-domain, then LCM-stability and reflexivity are equivalent for w-finite type torsion-free R-modules. Finally, we introduce the concept of w-LCM-stability for modules over a domain. Then we characterize when the module M is w-LCM-stable over the domain in terms of localizations and t-Nagata modules, respectively. Also we characterize Prufer v-multiplication domains and Krull domains in terms of w-LCM-stability.

Some Results on Gorenstein Dedekind Domains and Their Factor Rings

A domain is called a Gorenstein Dedekind domain (G-Dedekind for short) if every submodule of a projective module is G-projective (i.e., G-gldim(R) = 1). It is proved in this note that a domain R is a G-Dedekind domain if and only if every ideal of R is Gorenstein-projective (G-projective for short). We also show that nontrivial factor rings of Dedekind domains are QF-rings. We also give an example to show that factor rings of QF-rings are not necessarily QF-rings.

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

Spectra of Maximal 1-Sided Ideals and Primitive Ideals

R is any ring with identity. Let Spec r (R) (resp. Max r (R), Prim r (R)) denote the set of all right prime ideals (resp. all maximal right ideals, all right primitive ideals) of R and let U r (eR) = {P ∈ Spec r (R)| e ∉ P}. Let 𝒜 = ∪P∈Prim r (R) Spec r P (R), where Spec r P (R) = {Q ∈Spec r P (R)|P is the largest ideal contained in Q}. A ring is called right quasi-duo if every maximal right ideal is 2-sided. In this article, we study the properties of the weak Zariski topology on 𝒜 and the relationships among various ring-theoretic properties and topological conditions on it. Then the following results are obtained for any ring R: (1) R is right quasi-duo ring if and only if 𝒜 is a space with Zariski topology if and only if, for any Q ∈ 𝒜, Q is irreducible as a right ideal in R. (2) For any clopen (i.e., closed and open) set U in ℬ = Max r (R) ∪ Prim r (R) (resp. 𝒞 = Prim r (R)) there is an element e in R with e 2 − e ∈ J(R) such that U = U r (eR) ∩ ℬ (resp. U = U r (eR) ∩ 𝒞), where J(R) is the Jacobson of R. (3) Max r (R) ∪ Prim r (R) is connected if and only if Max l (R) ∪ Prim l (R) is connected if and only if Prim r (R) is connected.

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)