Prime Submodules Research Articles

MULTIPLICATION MODULES WHOSE ENDOMORPHISM RINGS ARE INTEGRAL DOMAINS

In this paper, several properties of endomorphism rings of modules are investigated. A multiplication module M over a commutative ring R induces a commutative ring <TEX>$M^*$</TEX> of endomorphisms of M and hence the relation between the prime (maximal) submodules of M and the prime (maximal) ideals of <TEX>$M^*$</TEX> can be found. In particular, two classes of ideals of <TEX>$M^*$</TEX> are discussed in this paper: one is of the form <TEX>$G_{M^*}\;(M,\;N)\;=\;\{f\;{\in}\;M^*\;|\;f(M)\;{\subseteq}\;N\}$</TEX> and the other is of the form <TEX>$G_{M^*}\;(N,\;0)\;=\;\{f\;{\in}\;M^*\;|\;f(N)\;=\;0\}$</TEX> for a submodule N of M.

Weakly unique factorization modules

In this work we give the definition of weakly prime element of a module. Therefore we give a new definition of factorization in a module, which is called weakly factorization. So we call a module weakly unique factorization module if all elements have a weakly factorization which is unique. We give the relation between weakly prime elements and weakly prime submodules. Then we characterize such weakly unique factorization modules.

Height of Prime and Weakly Prime Submodules

We will introduce and study new definitions for the height of prime submodules and weakly prime submodules. Also new module versions of catenary are discussed.

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.)

PRIMENESS IN MODULE CATEGORY

We introduce the notion of prime submodules of a given right R-module and describe all properties of them as a generalization of prime ideals in associative rings.

EQUIDIMENSIONAL SYMMETRIC ALGEBRAS

By using as main tool the theory of prime submodules this article is devoted to describing the structure of the minimal prime ideals of the equidimensional symmetric algebra of a finitely generated module.

PRIME SUBMODULES OF ARTINIAN MODULES

Prime submodules and weakly prime submodules of Artinian modules are characterized. Furthermore, some previous results on prime modules are generalized.

The Zariski Topology on the Prime Spectrum of a Module over Noncommutative Rings

Let R be an associative ring with identity and M an R-module. Let Spec (M) be the set of all prime submodules of M. We topologize Spec (M) with the Zariski topology and prove some useful results.

φ-PRIME SUBMODULES

AbstractLet R be a commutative ring with non-zero identity and M be a unitary R-module. Let (M) be the set of all submodules of M, and φ: (M) → (M) ∪ {∅} be a function. We say that a proper submodule P of M is a prime submodule relative to φ or φ-prime submodule if a ∈ R and x ∈ M, with ax ∈ P ∖ φ(P) implies that a ∈(P :RM) or x ∈ P. So if we take φ(N) = ∅ for each N ∈ (M), then a φ-prime submodule is exactly a prime submodule. Also if we consider φ(N) = {0} for each submodule N of M, then in this case a φ-prime submodule will be called a weak prime submodule. Some of the properties of this concept will be investigated. Some characterisations of φ-prime submodules will be given, and we show that under some assumptions prime submodules and φ1-prime submodules coincide.

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.

CLASSICAL PRIMARY SUBMODULES AND DECOMPOSITION THEORY OF MODULES

We introduce the notion of classical primary submodules that generalizes the concept of primary ideals of commutative rings to modules. Existence and uniqueness of classical primary decompositions in finitely generated modules over one-dimensional Noetherian domains are proved.

Weak Essential Submodules

A non-zero submodule N of M is called essential if N L for each non-zero submodule L of M. And a non-zero submodule K of M is called semi-essential if K P for each non-zero prime submodule P of M. In this paper we investigate a class of submodules that lies between essential submodules and semi-essential submodules, we call these class of submodules weak essential submodules.

Weak Essential Submodules

A non-zero submodule N of M is called essential if N L for each non-zero submodule L of M. And a non-zero submodule K of M is called semi-essential if K P for each non-zero prime submodule P of M. In this paper we investigate a class of submodules that lies between essential submodules and semi-essential submodules, we call these class of submodules weak essential submodules.

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 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 \( \sqrt[p]{N} \):= { m e M: every m-system containing m meets N}. It is shown that \( \sqrt[p]{N} \) is the intersection of all prime submodules of M containing N. We define radR(M) = \( \sqrt[p]{{(0)}} \). 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/radR(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/radR(M) is an Artinian R-module. This yields if M is an Artinian module over a PI-ring R, then either radR(M) = M or radR(M) = ∩i=1n\( \mathcal{P}_i M \) for some maximal ideals \( \mathcal{P}_1 , \ldots ,\mathcal{P}_n \) of R. Also, Baer’s lower nilradical of M [denoted by Nil* (RM)] is defined to be the set of all strongly nilpotent elements of M. It is shown that, for any projective R-module M, radR(M) = Nil*(RM) and, for any module M over a left Artinian ring R, radR(M) = Nil*(RM) = Rad(M) = Jac(R)M.

On the Maximal Spectrum of Semiprimitive Multiplication Modules

AbstractAnR-moduleMis called a multiplication module if for each submoduleNofM,N=IMfor some idealIofR. As defined for a commutative ringR, anR-moduleMis said to be semiprimitive if the intersection of maximal submodules ofMis zero. The maximal spectra of a semiprimitive multiplication moduleMare studied. The isolated points of Max(M) are characterized algebraically. The relationships among the maximal spectra ofM, Soc(M) and Ass(M) are studied. It is shown that Soc(M) is exactly the set of all elements ofMwhich belongs to every maximal submodule ofMexcept for a finite number. If Max(M) is infinite, Max(M) is a one-point compactification of a discrete space if and only ifMis Gelfand and for some maximal submoduleK, Soc(M) is the intersection of all prime submodules ofMcontained inK. WhenMis a semiprimitive Gelfand module, we prove that every intersection of essential submodules ofMis an essential submodule if and only if Max(M) is an almost discrete space. The set of uniform submodules ofMand the set of minimal submodules ofMcoincide. Ann(Soc(M))Mis a summand submodule ofMif and only if Max(M) is the union of two disjoint open subspacesAandN, whereAis almost discrete andNis dense in itself. In particular, Ann(Soc(M)) = Ann(M) if and only if Max(M) is almost discrete.

On Quasi Prime Fuzzy Submodules and Quasi Primary Fuzzy Submodules

In this paper we give some results about fuzzy quasi prime submodules, also,we study the notion of quasi primary fuzzy submodules

On weakly prime submodules

Let $ R $ be a commutative ring with non-zero identity. We define a proper submodule $ N $ of an $ R $-module $ M $ to be weakly prime if $ 0\not = rm\in N $( $ r\in R, m\in M $) implies $ m\in N $ or $ rM\subseteq N $. A number of results concerning weakly prime submodules are given. For example, we give three other characterizations of weakly 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).

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.