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).      

Reduced multiplication modules

An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. As defined for a commutative ring R, an R-module M is said to be reduced if the intersection of prime submodules of M is zero. The prime spectrum and minimal prime submodules of the reduced module M are studied. Essential submodules of M are characterized via a topological property. It is shown that the Goldie dimension of M is equal to the Souslin number of Spec(M)\mbox{\rm Spec}(M). Also a finitely generated module M is a Baer module if and only if Spec(M)\mbox{\rm Spec}(M) is an extremally disconnected space; if and only if it is a CS-module. It is proved that a prime submodule N is minimal in M if and only if for each x ∈ N, Ann(x) \not í (N:M).\mbox{\rm Ann}(x) \not \subseteq (N:M). When M is finitely generated; it is shown that every prime submodule of M is maximal if and only if M is a von Neumann regular module (VNM); i.e., every principal submodule of M is a summand submodule. Also if M is an injective R-module, then M is a VNM.     

Strongly Irreducible Submodules of Modules

Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

On Prime Submodules of a Finitely Generated Free Module Over a Commutative Ring

In this article, we give a full characterization of prime submodules of a finitely generated free module F over a commutative ring R with identity. Also we study the existence of primary decomposition of a submodule of F and characterize the minimal primary decomposition of this submodule. We also describe the structure of prime submodules of a module over a Dedekind domain.     

Prime and primary submodules of certain modules

In this paper we characterize all prime and primary submodules of the free R-module R ⁿ for a principal ideal domain R and find the minimal primary decomposition of any submodule of R ⁿ . In the case n = 2, we also determine the height of prime submodules.     

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.     

Weakly automorphism invariant modules and essential tightness

While a module is pseudo-injective if and only if it is automorphism-invariant, it was not known whether automorphism-invariant modules are tight. It is shown that weakly automorphism-invariant modules are precisely essentially tight. We give various examples of weakly automorphism-invariant and essentially tight modules and study their properties. Some particular results: (1) R is a semiprime right and left Goldie ring if and only if every right (left) ideal is weakly injective if and only if every right (left) ideal is weakly automorphism invariant; (2) R is a CEP-ring if and only if R is right artinian and every indecomposable projective right R-module is uniform and essentially R-tight.     

Weakly distributive modules. Applications to supplement submodules

In this paper, we define and study weakly distributive modules as a proper generalization of distributive modules.We prove that, weakly distributive supplemented modules are amply supplemented. In a weakly distributive supplemented module every submodule has a unique coclosure. This generalizes a result of Ganesan and Vanaja. We prove that π-projective duo modules, in particular commutative rings, are weakly distributive. Using this result we obtain that in a commutative ring supplements are unique. This generalizes a result of Camillo and Lima. We also prove that any weakly distributive ⊕-supplemented module is quasi-discrete.     

Strongly radical supplemented modules

Z?schinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.     

Strongly Prime Submodules

Let R be a commutative ring with identity. For an R-module M, the notion of strongly prime submodule of M is defined. It is shown that this notion of prime submodule inherits most of the essential properties of the usual notion of prime ideal. In particular, the Generalized Principal Ideal Theorem is extended to modules.     

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.     

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.     

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.     

Weakly regular modules over normal rings

Under study are some conditions for the weakly regular modules to be closed under direct sums and the rings over which all modules are weakly regular. For an arbitrary right R-module M, we prove that every module in the category σ(M) is weakly regular if and only if each module in σ(M) is either semisimple or contains a nonzero M-injective submodule. We describe the normal rings over which all modules are weakly regular.     

Abelian groups flat as modules over their endomorphism ring

Let R be a commutative ring with identity and let M be an R-module. We examine the situation where for each prime ideal ρof R the set of all ρ-prime submodules of M is finite. In case R is Noetherian and M is finitely generated, we prove that this condition is equivalent to there being a positive integer n such that for every prime ideal ρ of R, the number of ρ-prime submodules of Mis less than or equal to n. We further show that in this case, there is at most one ρ-prime submodule for all but finitely many prime ideals ρ of R.     

Some Remarks on Generalized GCD Domains

An integral domain R is a generalized GCD (GGCD) domain if the semigroup of invertible ideals of R is closed under intersection. In this article we extend the definition of PF-prime ideals to GGCD domains and develop a theory of these ideals which allows us to characterize Prüfer and π -domains among GGCD domains. We also introduce the concept of generalized GCD modules as a natural generalization of GGCD domains to the module case. An R-module M is a GGCD module if the set of invertible submodules of M is closed under intersection. We show that an integral domain R is a GGCD domain if and only if a faithful multiplication R-module M is a GGCD module. Various properties and characterizations of faithful multiplication GGCD modules over integral domains are considered and consequently, necessary and sufficient conditions for a ring R(M), the idealization of M, to be a GGCD ring are given.     

On generalization of ⊕-cofinitely supplemented modules

We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, cgs ^⊕-modules. It is shown that a module with summand sum property (SSP) is cgs ^⊕ if and only if M/w Loc^⊕ M (w Loc^⊕ M is the sum of all w-local direct summands of a module M) does not contain any maximal submodule, that every cofinite direct summand of a UC-extending cgs ^⊕-module is cgs ^⊕, and that, for any ring R, every free R-module is cgs ^⊕ if and only if R is semiperfect.     

Euclidean modules

In this paper, we introduce the notion of Euclidean module and weakly Euclidean ring. We prove the main result that a commutative ring R is weakly Euclidean if and only if every cyclic R-module is Euclidean, and also if and only if End( _R M) is weakly Euclidean for each cyclic R-moduleM. In addition, some properties of torsion-free Euclidean modules are presented.