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.     

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 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.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

Generalization of Nilpotency of Ring Elements to Module Elements

We define nilpotent and strongly nilpotent elements of a module M and show that the set 𝒩 _s (M) of all strongly nilpotent elements of M over a commutative unital ring R coincides with the classical prime radical β _cl (M) the intersection of all classical prime submodules of M.     

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.     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

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.     

t-Extending Modules and t-Baer Modules

We introduce the notions of “t-extending modules,” and “t-Baer modules,” which are generalizations of extending modules. The second notion is also a generalization of nonsingular Baer modules. We show that a homomorphic image (hence a direct summand) of a t-extending module and a direct summand of a t-Baer module inherits the property. It is shown that a module M is t-extending if and only if M is t-Baer and t-cononsingular. The rings for which every free right module is t-extending are called right Σ-t-extending. The class of right Σ-t-extending rings properly contains the class of right Σ-extending rings. Among other equivalent conditions for such rings, it is shown that a ring R is right Σ-t-extending, if and only if, every right R-module is t-extending, if and only if, every right R-module is t-Baer, if and only if, every nonsingular right R-module is projective. Moreover, it is proved that for a ring R, every free right R-module is t-Baer if and only if Z ₂(R _R ) is a direct summand of R and every submodule of a direct product of nonsingular projective R-modules is projective.     

Comparing First Order Theories of Modules over Group Rings II: Decidability

We consider R‐torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. In the first part of the paper [4] we compared the theory T(RG) of all R‐torsionfree RG‐modules and the theory T₀(RG) of RG‐lattices (i. e. finitely generated R‐torsionfree RG‐modules), and we realized that they are almost always different. Now we compare their behaviour with respect to decidability, when RG‐lattices are of finite, or wild representation type.     

The genus of a module

Let R be a Dedekind domain satisfying the Jordan-Zassenhaus theorem (e.g., the ring of integers in a number field) and Λ a module finite R-algebra. We extend classical results of Jacobinski, Roiter, and Drozd on orders and lattices. In particular, it is shown that the genus of a finitely generated Λ-module M is finite. Moreover, given M, there exist a positive integer t and a finite extension S of R such that a Λ-module N is the genus of M if and only if M^(t) ? N^(t) if and only if M ? S ? N ? S.     

Modules and Spectral Spaces

We establish conditions for Spec(M) to be Noetherian and spectral space, w.r.t. different topologies. We used rings with Noetherian spectrum to produce plentiful examples of modules with Noetherian spectrum that have not appeared in the literature previously. In particular, we show that every ?-module has Noetherian spectrum. Another main subject of this article is presenting the conditions under which a module is top. In particular, we show that every distributive module is top, every content weak multiplication R-module M is also top, and moreover, if R has Noetherian spectrum, then Spec(M) is a spectral space.     

Injectivity relative to closed submodules

Let R be a ring. An R-module X is called c-injective if, for every closed submodule L of every R-module M, every homomorphism from L to X lifts to M. It is proved that if R is a Dedekind domain then an R-module X is c-injective if and only if X is isomorphic to a direct product of homogeneous semisimple R-modules and injective R-modules. It is also proved that a commutative Noetherian domain R is Dedekind if and only if every simple R-module is c-injective.     

Catenary Modules

We generalise the concept of catenary rings to modules. We call an A-module M catenary if for each pair of prime submodules K and L of M with K ⊂ L all saturated chains of prime submodules of M from K to L have a common finite length. We show that any finitely generated module over a PID is catenary and also being catenary is a local property. Moreover, we prove that when A is a one dimensional Noetherian domain, then A is a Dedekind domain if and only if every finitely generated torsion-free A-module is catenary. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some new dimensions of modules and rings

We say that a class 𝒫 of right modules over a fixed ring R is an epic class if it is closed under homomorphic images. For an arbitrary epic class 𝒫, we define a 𝒫-dimension of modules that measures how far modules are from the modules in the class 𝒫. For an epic class 𝒫 consisting of indecomposable modules, first we characterize rings whose modules have 𝒫-dimension. In fact, we show that every right R-module has 𝒫-dimension if and only if R is a semisimple Artinan ring. Then we study fully Hopfian modules with 𝒫-dimension. In particular, we show that a commutative ring R with 𝒫-dimension (resp. finite 𝒫-dimension) is either local or Noetherian (resp. Artinian). Finally, we show that Mat_m(R) is a right Köthe ring for some m if and only if every (left) right module is a direct sum of modules of 𝒫-dimension at most n for some n, if and only if R is a pure semisimple ring.     

On the prime radical and Baer’s lower nilradical of modules

Let M be a left R-module. In this paper a generalization of the notion of m-system set of rings to modules is given. Then for a submodule N of M, we define := { m ε M: every m-system containing m meets N}. It is shown that is the intersection of all prime submodules of M containing N. We define rad _R (M) = . This is called Baer-McCoy radical or prime radical of M. It is shown that if M is an Artinian module over a PI-ring (or an FBN-ring) R, then M/rad _R (M) is a Noetherian R-module. Also, if M is a Noetherian module over a PI-ring (or an FBN-ring) R such that every prime submodule of M is virtually maximal, then M/rad _R (M) is an Artinian R-module. This yields if M is an Artinian module over a PI-ring R, then either rad _R (M) = M or rad _R (M) = ∩ _i=1 ⁿ for some maximal ideals of R. Also, Baer’s lower nilradical of M [denoted by Nil_* ( _R M)] is defined to be the set of all strongly nilpotent elements of M. It is shown that, for any projective R-module M, rad _R (M) = Nil_*( _R M) and, for any module M over a left Artinian ring R, rad _R (M) = Nil_*( _R M) = Rad(M) = Jac(R)M. This research was in part supported by a grant from IPM (No. 85130016). Also this work was partially supported by IUT (CEAMA). The author would like to thank the anonymous referee for a careful checking of the details and for helpful comments that improved this paper.     

Properties of a certain product of submodules

Let R be a commutative ring with identity, let M be an R-module, and let K ₁, . . . ,K _n be submodules of M: We construct an algebraic object called the product of K ₁, . . . ,K _n : This structure is equipped with appropriate operations to get an R(M)-module. It is shown that the R(M)-module M ⁿ   = M . . .M and the R-module M inherit some of the most important properties of each other. Thus, it is shown that M is a projective (flat) R-module if and only if M ⁿ is a projective (flat) R(M)-module.     

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.