Unitary R-module Research Articles

Non-torsion element graph of a module over a commutative ring

Let R be a commutative ring with unity and M be a unitary R-module. Let T(M) be the set of all torsion elements of M and NT(M) = M − T(M) be the set of all non-torsion elements of M. The non-torsion element graph of M over R is an undirected simple graph GNT (M) with NT(M) as vertex set and any two distinct vertices x and y are adjacent if and only if x + y ∈ T(M). In this paper, we study the basic properties of the graph GNT (M). We also study the diameter and girth of GNT (M). Further, we determine the domination number and the bondage number of GNT (M). We establish a relation between diameter and domination number of GNT (M). We also establish a relation between girth and bondage number of GNT (M). We also establish a relation between girth and bondage number of GNT (M).

Closed (St-Closed) Compressible Modules and Closed (St-Closed) Retractable Modules

Let be a commutative ring with 1 and be a left unitary R-module. In this paper, we give a generalization for the notions of compressible (retractable) module. As well as, we study closed (St-closed) compressible and closed (St-closed) retractable. Furthermore, some of their advantages, properties, categorizations and instances have been given. Finally, we study the relation between them.

Some Properties of S-Semiannihilator Small Submodules and S-Small Submodules with respect to a Submodule

Let R be a commutative ring with nonzero identity, S⊆R be a multiplicatively closed subset of R, and M be a unital R-module. In this article, we introduce the concepts of S-semiannihilator small submodules and S-T-small submodules as generalizations of S-small submodules. We investigate some basic properties of them and give some characterizations of such submodules, especially for (finitely generated faithful) multiplication modules.

S-Essentially Compressible Modulesand S-Essentially Retractable Modules

Let R be a commutative ring with 1 and M be a left unitary R-module. In this paper, we give a generalization for the notions of compressible (retractable) Modules. We study s-essentially compressible (s-essentially retractable). We give some of their advantages, properties, characterizations and examples. We also study the relation between s-essentially compressible (s-essentially retractable modules) and some classes of modules.

On domination in the total torsion element graph of a module

Let R be a commutative ring with non-zero unity and M be a unitary R-module. Let T(M) be the set of torsion elements of M. Atani and Habibi [6] introduced the total torsion element graph of M over R as an undirected graph T(Γ(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x + y ∈ T(M). The main objective of this paper is to study the domination properties of the graph T(Γ(M)). The domination number of T(Γ(M)) and its induced subgraphs T or(Γ(M)) and T of(Γ(M)) has been determined. Some domination parameters of T(Γ(M)) are also studied. In particular, the bondage number of T(Γ(M)) has been determined. Finally, it has been proved that T(Γ(M)) is excellent, domatically full and well covered under certain conditions.

On Weakly S-2-Absorbing Submodules

Let R be a commutative ring with identity and let M be a unitary R-module. In this paper, we introduce the notion of weakly S-2-absorbing submodule. Suppose that S is a multiplicatively closed subset of R. A submodule P of M with (P:R M)∩S=∅ is said to be a weakly S-2-absorbing submodule if there exists an element s ∈ S such that whenever a,b∈R and m∈M with 0≠abm∈P, then sab∈(P: M) or sam∈P or sbm∈P. We give the characterizations, properties and examples of weakly S-2-absorbing submodules.

Quasi–essentially pseudo – prime modules

Let R be a commutative ring with identity and M be a unitary R-module. In this paper we introduce the concept Quasi-essentially pseudo-prime module as a generalization of a prime module and give of examples, characterizations and some basic properties of this concept. Further more we study the relationships of Quasi-essentially pseudo-prime modules with some classes of modules .

Nearly 2-Absorbing Submodules And Related Concepts

Throughout this note R is commutative ring with identity, and X be aleft unitary R-module. A proper submodule K of an R-module X is callednearly prime, if whenever implies that either( ) or [ ( ) ]. This concept us to introduce theconcept of nearly prime submodule, where a proper submodule K of anR-module X is called nearly 2-absorbing submodule, if wherever, and then either ( ) or( ) or [ ( ) ]. The aim of this paper is to study thisconcept, and gives some of its basic properties, characterization andexamples. Furthermore, we study the relation of this concept with somekind of submodules".

On weakly S-primary submodules

Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N :R M) ? S = ? to be weakly S-primary if there exists s ? S such that whenever a ? R and m ? M with 0 , am ? N, then either sa ??(N :R M) or sm ? N. We present various properties and characterizations of this concept (especially in faithful multiplication modules). Moreover, the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations is investigated. Finally, we determine some conditions under which two kinds of submodules of the amalgamation module along an ideal are weakly S-primary.

Zariski topology on the spectrum of fuzzy classical primary submodules

Let R be a commutative ring with identity and M a unitary R-module. The fuzzy classical primary spectrum F cp.spec(M) is the collection of all fuzzy classical primary submodules A of M, the recent generalization of fuzzy primary ideals and fuzzy classical prime submodules. In this paper, we topologize FM(M) with a topology having the fuzzy primary Zariski topology on the fuzzy classical primary spectrum F cp.spec(M) as a subspace topology, and investigate the properties of this topological space.

2-prime submodules of modules

Let R be a commutative ring with unity. And let E be a unitary R-module. This paper introduces the notion of 2-prime submodules as a generalized concept of 2-prime ideal, where proper submodule H of module F over a ring R is said to be 2-prime if , for r R and x F implies that or . we prove many properties for this kind of submodules, Let H is a submodule of module F over a ring R then H is a 2-prime submodule if and only if [N ] is a 2-prime submodule of E, where r R. Also, we prove that if F is a non-zero multiplication module, then [K: F] [H: F] for every submodule k of F such that H K. Furthermore, we will study the basic properties of this kind of submodules.

On graded 1-absorbing prime submodules

Let G be a group with identity e, R be a commutative G-graded ring with unity 1 and M be a G-graded unital R-module. In this article, we introduce the concept of graded 1-absorbing prime submodule. A proper graded R-submodule N of M is said to be a graded 1-absorbing prime R-submodule of M if for all non-unit homogeneous elements x, y of R and homogeneous element m of M with xym ∈ N, either xy ∈ (N :R M) or m ∈ N. We show that the new concept is a generalization of graded prime submodules. At the same time it is a special graded 2-absorbing submodule. Several properties of a graded 1-absorbing prime submodule have been obtained. We investigate graded 1-absorbing prime submodules when the components {Mg : g ∈ G} are multiplication Re -modules.

Essential ideals represented by mod-annihilators of modules

Let R be a commutative ring with unity, M be a unitary R-module and G a finite abelian group (viewed as a \({\mathbb {Z}}\)-module). The main objective of this paper is to study properties of mod-annihilators of M. For \(x \in M\), we study the ideals \([x : M] =\{r\in R ~|~ rM\subseteq Rx\}\) of R corresponding to mod-annihilator of M. We investigate as when [x : M] is an essential ideal of R. We prove that the arbitrary intersection of essential ideals represented by mod-annihilators is an essential ideal. We observe that [x : M] is injective if and only if R is non-singular and the radical of R/[x : M] is zero. Moreover, if essential socle of M is non-zero, then we show that [x : M] is the intersection of maximal ideals and \([x : M]^2 = [x : M]\). Finally, we discuss the correspondence of essential ideals of R and vertices of the annihilating graphs realized by M over R.

Proof of Cayley-Hamilton theorem using polynomials over the algebra of module endomorphisms

If R is a commutative unital ring and M is a unital R-module, then each element of EndR(M) determines a left EndR(M)[X]-module structure on EndR(M), where EndR(M) is the R-algebra of endomorphisms of M and EndR(M)[X]=EndR(M)⊗RR[X]. These structures provide a very short proof of the Cayley-Hamilton theorem, which may be viewed as a reformulation of the proof in Algebra by Serge Lang. Some generalisations of the Cayley-Hamilton theorem can be easily proved using the proposed method.

CHARACTERIZATION OF WEAKLY PRIME SUBMODULE

Let R is a commutative ring with identity and M is a unital R-module. El-bast and Smith (1988) have researched and introduced the multiplication module. Prime submodule has been studied by Ameri (2002). Then Atani and Farzalipour (2007) have extended the prime submodule to weakly prime submodule. Some properties were proved on each paper. This paper will study about characterization of weakly prime submodules.

1-absorbing primary submodules

Abstract Let R be a commutative ring with non-zero identity and M be a unitary R-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule N of M is said to be a 1-absorbing primary submodule if whenever non-unit elements a, b ∈ R and m ∈ M with abm ∈ N, then either ab ∈ (N : RM) or m ∈ M − rad(N). Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.

A new topology over the primary-like spectrum of a module

<p>Let R be a commutative ring with identity and M a unitary R-module. The primary-like spectrum Spec<sub>L</sub>(M) is the collection of all primary-like submodules Q of M, the recent generalization of primary ideals, such that M/Q is a primeful R-module. In this article, we topologies Spec<sub>L</sub>(M) with the patch-like topology, and show that when, Spec<sub>L</sub>(M) with the patch-like topology is a quasi-compact, Hausdorff, totally disconnected space.</p>

Essentially Retractable Modules Relative To A Submodule And Some Generalizations

R is a ring with unity, and all modules are unitary right R-modules. The concept of compressible modules was introduced in 1981 by Zelmanowitz, where module M is called compressible if it can be embedded in any nonzero submodule A of M . In other words, M is a compressible module if for each nonzero submodule A of M, f 2 Hom(M;A) exists, such that f is monomorphism. Retractable modules were introduced in 1979 Khuri, where module M is retractable if Hom(M, A ) 6= 0 for every nonzero submodule A of M . We define a new notion, namely, essentially retractable module relative to a submodule. In addition, new generalizations of compressible modules relative to a submodule are introduced, where module M is called compressible module relative to a submodule N of M . If for all nonzero submodule K of M contains N , then a monomorphism f 2 Hom(M, K) exists. Some basic properties are studied and many relationships between these classes and other related concepts are presented and studied. We also introduce another generalization of retractable module, which is called small kernel retractable module

J-Small Semiprime Submodules

Let R be a commutative ring with identity and Y be an unitary R-module. We say a non-zero submodule s of Y is a J – small semiprime if and only if for whenever i ∈ R, y ∈ Y,(Y) is small in Y and i2y ∈ S + Rad (Y) implies iy ∈ S. In this paper, we investigate some properties and characterizations of these class of submodules

On weakly $$\delta _{R,M}$$-semiprimary submodules

Let R be a commutative ring with $$1\ne 0$$ and M be a nonzero unital R-module. Recall that a proper submodule N of M is called a semiprimary submodule of M if whenever $$rm\in N$$ for $$r\in R$$ and $$m \in M$$ , then $$r\in \sqrt{N:M}$$ or $$m\in \sqrt{N}$$ , where $$rad(N)=\sqrt{N}$$ is the intersection of all prime submodules of M containing N. We define a proper submodule N of M to be a weakly semiprimary submodule if whenever $$0_{M}\ne rm\in N$$ for $$r\in R$$ and $$m \in M$$ , then $$r\in \sqrt{N:M}$$ or $$m\in \sqrt{N}$$ . In this paper, we give a characterization of generalizations of (weakly) semiprimary submodules. Let S(M) be the set of all submodules of M and $$\delta _{M}:S(M)\rightarrow S(M)$$ be a function. Then we say $$\delta _{M}$$ is an expansion of submodules of M if whenever A, B, C are submodules of M with $$A\subseteq B$$ , then $$C\subseteq \delta _{M}(C)$$ and $$\delta _{M}(A)\subseteq \delta _{M}(B)$$ . Let $$\delta _{M}$$ be an expansion of submodules and $$\delta _{R}$$ be an expansion of ideals. Then a proper submodule N is called a ( $$\delta _{R,M}$$ -semiprimary) weakly $$\delta _{R,M}$$ -semiprimary submodule of M if $$(rm\in N)$$ $$0_{M}\ne rm\in N$$ implies $$r\in \delta _{R}(N:M)$$ or $$m\in \delta _{M}(N)$$ . Various results and examples concerning weakly $$\delta _{R,M}$$ -semiprimary submodules are given.