On prime submodules and primary decomposition

We characterize prime submodules of R × R for a principal ideal domain R and investigate the primary decomposition of any submodule into primary submodules of R × R.     

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.     

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.     

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 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).      

Saturations of Submodules

Abstract

Let N be a submodule of a module M over a commutative ring R such that M/N is finitely generated. It is shown that a submodule of M is a prime submodule minimal over N if and only if it is the saturation of N + pM for certain prime ideal p of R. The bearing of this result upon the M-radical of N is discussed.     

PRIME AND RADICAL SUBMODULES OF MODULES OVER COMMUTATIVE RINGS

ABSTRACT

In this paper, unless otherwise stated, all rings are commutative with identity and all modules are unital. We give sufficient conditions to ensure that a submodule has a module-reduced primary decomposition. In general, the radical of a primary submodule is not prime and the radical does not split intersections of submodules, as is valid in the ideal case. We study sufficient conditions for which these properties hold in the module setting. These conditions involve dimension arguments, consideration of finitely generated modules, and the spectrum of a given prime ideal. Further, we consider the computational problem of finding a Gröbner basis of both the colon and the radical of a submodule. A characterization of the elements of the colon is given, along with a method of computing the radical of a submodule in certain cases.     

On Strongly Essential Submodules

The submodules with the property of the title (N ? M is strongly essential in M if ∏ _I N is essential in ∏ _I M for any index set I) are introduced and fully investigated.

It is shown that for each submodule N of M there exists a subset T ? M such that N + T is strongly essential submodule of M and (N:T) = Ann(T), T  ∩  N = 0. Basic properties of these objects and several examples are given and the counterparts of the related concepts to essential submodules are also introduced and studied. It is shown that each maximal left ideal of a left fully bounded ring is either a summand or strongly essential. Rings over which no module has a proper strongly essential submodule are characterized. It is also shown that the left Loewy rings are the only rings over which the essential submodules and strongly essential submodules of any left module coincide. Finally, a new characterization of left FBN rings is observed.     

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.     

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.     

RINGS WHOSE SIMPLE MODULES ARE ABSOLUTELY PURE

A ring is called right SAP if every right simple module over it is absolutely pure. In this paper we prove that every right SAP ring is semiprimitive and that the homomorphic image and the center of an right SAP ring are also right SAP. We also show that the sum of all absolutely pure minimal submodules of any module is a fully invariant submodule. As an application, we give a decomposition of some selfinjective rings.     

Uniformly Classical Primary Submodules

We shall introduce the notion of uniformly classical primary submodule that generalizes the concept of uniformly primary ideal as given by J. A. Cox and A. J. Hetzel. We also advance the companion concepts of fully uniformly classical primary module and uniformly primary compatible module. Along these lines, we present a characterization of Noetherian rings R for which every R-module is fully uniformly classical primary and we present a characterization of rings R for which every finitely generated R-module is uniformly primary compatible. Results illustrating connections among the notions of uniformly classical primary submodule, uniformly primary ideal, and uniformly primary submodule as given by R. Ebrahimi-Atani and S. Ebrahimi-Atani are also provided.     

Projective modules and prime submodules

In this paper, we use Zorn’s Lemma, multiplicatively closed subsets and saturated closed subsets for the following two topics (i) The existence of prime submodules in some cases (ii) The proof that submodules with a certain property satisfy the radical formula. We also give a partial characterization of a submodule of a projective module which satisfies the prime property.     

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 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.     

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.     

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.     

Radical of submodules and symmetric algebra

A prime ideal of the symmetric algebra of an R-module N is associated to every prime submodule M ?N and this assignment is then used to obtain a new characterization of the radical of a submodule. Several applications of these results are also included.     

Modules with Noetherian Spectrum

Let M be a module over a commutative ring R. A submodule P of M is called prime if P ≠ M and, whenever r ∈ R, e ∈ M, and re ∈ P, we have rM ? P or e ∈ P. We let Spec(M) denote the set of all prime submodules of M. Using a topology analogous to the Zariski topology for Spec(R), we establish necessary and sufficient conditions for Spec(M) to be a Noetherian space. We produce some examples of modules with Noetherian spectrum that have not appeared in the literature previously. In particular, Laskerian modules and faithfully flat modules over Laskerian rings have Noetherian spectra. (The term Laskerian is defined in Section 3.)     

Singular and nonsingular modules relative to a torsion theory

In this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept of τ-essential submodule for a hereditary torsion theory τ. We introduce two new module classes called τ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization of τ-semisimple rings via non-τ-singular modules. We prove that if M∕τ(M) is non-τ-singular for a module M, then every submodule of M has a unique τ-closure. We give some properties of the torsion theory generated by the class of all τ-singular modules. We obtain a decomposition theorem for a strongly τ-extending module by using non-τ-singular modules.