Idealization and Theorems of D. D. Anderson

All rings are commutative with identity and all modules are unital. Anderson proved that a submodule N of an R-module M is multiplication (resp. join principal) if and only if 0₍₊₎ N is a multiplication (resp. join principal) ideal or R(M). The idealization of M. In this article we develop more fully the tool of idealization of a module, particularly in the context of multiplication modules, generalizing Anderson's theorems and discussing the behavior under idealization of some ideals and some submodules associated with a module.     

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.     

On weakly prime radical of modules and semi-compatible modules

Summary Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN ⊆ P, we have AN ⊆ P or BN ⊆ P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/\B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module R⊕ R is a semi-compatible module, then R is a Bezout domain.     

The subring generated by the units in a compact ring

ABSTRACT

In this paper, the authors introduce the concept of integrally closed modules and characterize Dedekind modules and Dedekind domains. They also show that a given domain R is integrally closed if and only if a finitely generated torsion-free projective R-module is integrally closed. In addition, it is proved that any invertible submodule of a finitely generated projective module over a domain is finitely generated and projective. Also they give the equivalent conditions for Dedekind modules and Dedekind domains.

     

On the Prime Spectrum of X-Injective Modules

Let R be a commutative ring and M an R-module. The purpose of this article is to introduce a new class of modules over R called X-injective R-modules, where X is the prime spectrum of M. This class contains the family of top modules and that of weak multiplication modules properly. In this article our concern is to extend the properties of multiplication, weak multiplication, and top modules to this new class of modules. Furthermore, for a top module M, we study some conditions under which the prime spectrum of M is a spectral space for its Zariski topology.     

On Factorization in Modules

In two articles, Anderson and Valdes-Leon generalized the theory of factorization in integral domains to commutative rings with zero divisors and to modules. Here we investigate some factorization properties in modules and state a result that relates factorization properties of an R-module, M, to the factorization properties of M as an (R/Ann(M))-module. Furthermore, we will investigate when a polynomial module, M[x], has the bounded factorization property, assuming that M has this property.     

Some remarks on multiplication ideals,ii

Let R bea commutative ring with identity. An R-module (ideal of R) A is called a multiplication module (ideal) if for each submodule N of A there exists an ideal I of R with N = I A. We give several characterizations of multiplication modules. Using the method of idealization we show how to reduce questions concerning multiplication modules to multiplication ideals. For example, we show that if S is a commutative R-algebra and ψ: M→an R-module homomorphism where M is a multiplication R-module and N is an S-module, then Sψ(M) is a multiplication S-module.     

Finitely Presented Modules Over Semihereditary Rings

We prove that each almost local-global semihereditary ring R has the stacked bases property and is almost Bézout. More precisely, if M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annihilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a projective module which is isomorphic to a direct sum of finitely generated ideals. These ideals allow us to define a finitely generated ideal whose isomorphism class is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too. It follows that each semihereditary ring of Krull-dimension one or of finite character, in particular each hereditary ring, has the stacked base property. These results were already proved for Prüfer domains by Brewer, Katz, Klinger, Levy, and Ullery. It is also shown that every semihereditary Bézout ring of countable character is an elementary divisor ring.     

Another Gorenstein analogue of flat modules

A right R-module M is called glat if any homomorphism from any finitely presented right R-module to M factors through a finitely presented Gorenstein projective right R-module. The concept of glat modules may be viewed as another Gorenstein analogue of flat modules. We first prove that the class of glat right R-modules is closed under direct sums, direct limits, pure quotients and pure submodules for arbitrary ring R. Then we obtain that a right R-module M is glat if and only if M is a direct limit of finitely presented Gorenstein projective right R-modules. In addition, we explore the relationships between glat modules and Gorenstein flat (Gorenstein projective) modules. Finally we investigate the existence of preenvelopes and precovers by glat and finitely presented Gorenstein projective modules.     

On ADS Modules and Rings

A right module M over a ring R is said to be ADS if for every decomposition M = S ⊕ T and every complement T′ of S, we have M = S ⊕ T′. In this article, we study and provide several new characterizations of this new class of modules. We prove that M is semisimple if and only if every module in σ[M] is ADS. SC and SI rings also characterized by the ADS notion. A ring R is right SC-ring if and only if every 2-generated singular R-module is ADS.     

Artinness of Generalized Macaulay–Northcott Modules

Let R be an associated ring not necessarily with identity, M a left R-module having the property (F), and (S, ≤) a strictly totally ordered monoid which is also artinian and finitely generated. It is shown that the module [M ^S,≤] consisting of generalized inverse polynomials over M is an artinian left [[R ^S,≤]]-module if and only if M is an artinian left R-module.     

Some Remarks on Multiplication and Projective Modules II

All rings are commutative with identity and all modules are unital. Let R be a ring and M an R-module. In our recent work [6 Ali , M. M. , Smith D. J. ( 2004 ). Some remarks on multiplication and projective modules . Communications in Algebra 32 : 3897 – 3909 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] we investigated faithful multiplication modules and the properties they have in common with projective modules. In this article, we continue our study and investigate faithful multiplication and locally cyclic projective modules and give several properties for them. If M is either faithful multiplication or locally cyclic projective then M is locally either zero or isomorphic to R. We show that, if M is a faithful multiplication module or a locally cyclic projective module, then for every submodule N of M there exists a unique ideal Γ(N) ? Tr(M) such that N = Γ(N)M. We use this result to show that the structure of submodules of a faithful multplication or locally cyclic projective module and their traces are closely related. We also use the trace of locally cyclic projective modules to study their endomorphisms.     

On finiteness of multiplication modules

Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).      

On Dedekind Modules

In this article the authors give the relation between a finitely-generated torsionfree Dedekind module M over a domain R and prime submodules of the 𝒪(M)-module M and the ring 𝒪(M). They also prove that M is a finitely-generated torsionfree Dedekind module over a domain R if and only if every semi-maximal submodule of R-module M is invertible.     

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.     

Strongly Duo Modules and Rings

An R-module M is called strongly duo if Tr(N, M) = N for every N ≤ M _R . Several equivalent conditions to being strongly duo are given. If M _R is strongly duo and reduced, then End _R (M) is a strongly regular ring and the converse is true when R is a Dedekind domain and M _R is torsion. Over certain rings, nonsingular strongly duo modules are precisely regular duo modules. If R is a Dedekind domain, then M _R is strongly duo if and only if either M ≈ R or M _R is torsion and duo. Over a commutative ring, strongly duo modules are precisely pq-injective duo modules and every projective strongly duo module is a multiplication module. A ring R is called right strongly duo if R _R is strongly duo. Strongly regular rings are precisely reduced (right) strongly duo rings. A ring R is Noetherian and all of its factor rings are right strongly duo if and only if R is a serial Artinian right duo ring.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Idempotent and Nilpotent Submodules of Multiplication Modules

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.     

On Invertible and Dense Submodules

Abstract

In this paper, the authors give a partial characterization of invertible, dense and projective submodules. In the final section, they give the equivalent conditions to be invertible, dense and projective submodules for a given an R-module M. They also provide conditions under which a given ring R is a Dedekind domain if and only if every non zero submodule of an R-module is locally free.     

Infinite Direct Sums of Lifting Modules

A module M over a ring R is called a lifting module if every submodule A of M contains a direct summand K of M such that A/K is a small submodule of M/K (e.g., local modules are lifting). It is known that a (finite) direct sum of lifting modules need not be lifting. We prove that R is right Noetherian and indecomposable injective right R-modules are hollow if and only if every injective right R-module is a direct sum of lifting modules. We also discuss the case when an infinite direct sum of finitely generated modules containing its radical as a small submodule is lifting.