Prime Submodules Research Articles

On graded 2-classical prime submodules of graded modules over graded commutative rings

Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper, we introduce the concept of graded 2-classical prime submodules. Various properties of graded 2-classical prime submodules are considered.

I-Semiprime Submodules

Let be a commutative ring with identity and a fixed ideal of and be an unitary -module.We say that a proper submodule of is -semi prime submodule if with . In this paper, we investigate some properties of this class of submodules. Also, some characterizations of -semiprime submodules will be given, and we show that under some assumptions -semiprime submodules and semiprime submodules are coincided.

Approximaitly Semi-Prime Submodules and Some Related Concepts

We introduce in this paper the concept of approximaitly semi-prime submodules of unitary left -module over a commutative ring with identity as a generalization of a prime submodules and semi-prime submodules, also generalization of quasi-prime submodules and approximaitly prime submodules. Various basic properties of an approximaitly semi-prime submodules are discussed, where a proper submodule of an -module is called an approximaitly semi-prime submodule of , if whenever , where , and , implies that . Furthermore the behaviors of approximaitly semi-prime submodule in some classes of modules are studied. On the other hand several characterizations of this concept are introduced.

Chains and unions of prime submodules

Abstract. Let R be a commutative ring with identity and let M be a unital R-module. In this paper we study the various properties of prime submodules. Also we give a new equivalent conditions for a minimal prime submodules of an R-module to be a finite set, whenever R is a Noetherian ring. Finally we prove the Prime avoidance Theorem for modules in different states.

Approximaitly Primary Submodules

The study deals with the notion of an approximaitly primary submodules of unitary left -module over a commutative ring with identity as a generalization of a primary submodules and approximaitly prime submodules, where a proper submodule of an -module is called an approximaitly primary submodule of , if whenever , for , , implies that either or for some positive integer of . Several characterizations, basic properties of this concept are given. On the other hand the relationships of this concept with some classes of modules are studied. Furthermore, the behavior of approximaitly primary submodule under -homomorphism are discussed.

S(0)-Weakly second submodules

We introduce the dual notions of [Formula: see text]-weakly prime submodules, that is, [Formula: see text]-weakly second submodules in a commutative ring with identity. We investigate the properties of [Formula: see text]-weakly second submodules and obtain some related useful characterizations of this dualization. Moreover, we also prove some properties of [Formula: see text]-weakly second submodules related to multiplication and comultiplication modules.

On Beta-Prime Submodules

We introduce the concepts of $\beta$-prime submodules and weakly $\beta$-prime submodules of unital left modules over a commutative ring with nonzero identity. Some properties of these concepts are investigated. We use the notion of the product of two submodules to characterize $\beta$-prime submodules of a multiplication module. Characterization of $\beta$-prime and weakly $\beta$-prime submodules of arbitary modules are also given.

The set of strong torsion elements of a module over a noncommutative ring

Let [Formula: see text] be an arbitrary ring and [Formula: see text] be a nonzero right [Formula: see text]-module. In this paper, we introduce the set [Formula: see text], for some nonzero ideal [Formula: see text] of [Formula: see text] of strong torsion elements of [Formula: see text] and the properties of this set are investigated. In particular, we are interested when [Formula: see text] is a submodule of [Formula: see text] and when it is a union of prime submodules of [Formula: see text].

Almost prime and weakly prime submodules

Let [Formula: see text] be an arbitrary ring and [Formula: see text] be a right [Formula: see text]-module. A proper submodule [Formula: see text] of [Formula: see text] is called almost prime (respectively, weakly prime) if for each submodule [Formula: see text] of [Formula: see text] and each ideal [Formula: see text] of [Formula: see text] that [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] or [Formula: see text]. We study these notions which are new generalizations of the prime submodules over noncommutative rings and we obtain some related results. We show that these two concepts in some classes of modules coincide. Moreover, we investigate the conditions that [Formula: see text] is almost prime, where [Formula: see text] is a submodule of [Formula: see text] and [Formula: see text] is an ideal of [Formula: see text]. Also, the almost prime radical of modules will be introduced and we extend some known results.

Approximaitly Prime Submodules and Some Related Concepts

In this research note approximately prime submodules is defined as a new generalization of prime submodules of unitary modules over a commutative ring with identity. A proper submodule of an -module is called an approximaitly prime submodule of (for short app-prime submodule), if when ever , where , , implies that either or . So, an ideal of a ring is called app-prime ideal of if is an app-prime submodule of -module . Several basic properties, characterizations and examples of approximaitly prime submodules were given. Furthermore, the definition of approximaitly prime radical of submodules of modules were introduced, and some of it is properties were established.

Prime virtually semisimple modules and rings

This article is a sequel to the recent three papers on “virtually semisimple modules and rings,” by Behboodi et al., which two of them appeared in the Algebras and Representation Theory and Communications in Algebra in 2018. An R-module M is called virtually semisimple if each submodule of M is isomorphic to a direct summand. A ring R is called left (resp., right) virtually semisimple if (resp., RR) is virtually semisimple. In this article, we study rings and modules in which every prime submodule is isomorphic to a direct summand, and called them prime virtually (or -virtually) semisimple modules. A ring R is called left (resp., right) -virtually semisimple if (resp., RR) is -virtually semisimple. The results of the article are inspired by a characterization of left -virtually semisimple rings. We prove that these rings are precisely the left virtually semisimple rings, and in this case , where each Di is a domain and each is a principal left ideal ring. We also answer to the following questions: (i) Describe rings R where each (finitely generated or cyclic) left R-module is -virtually semisimple?, and (ii) Describe rings R where each left R-module is a direct sum of indecomposable -virtually semisimple modules? Finally, we study -virtually semisimple modules over commutative rings.

WE-primary submodules and WE-quasi-prime submodules

In this paper we introduced and study two concepts one of is a subclass of a class of weakly primary submodules called WE-primary submodules and the other of a subclass of weakly quasi-prime submodules called WE-quasi-prime submodules. Several properties of WE-primary and WE- quasi - prime submodules have been given, and some characterizations of each type of submodules are introduced moreover behavior of these submodules in some classes of modules are investigated. On the otherhand, the homeorphic and inverse image of these concept under certain conditions are proved

On almost fuzzy prime ideals and sub-modules

The purpose of this paper is to present some characterizations of almost fuzzy prime ideals and sub modules; and to introduce certain innovative fuzzy prime ideals and sub modules respectively. Moreover, every fuzzy ideal in R is an almost prime ideal for a quasi-local ring \( (R, M) \) with \( M^{2} = 0 \). Moreover, prime sub module µN in a finitely generated multiplication R − module M and fuzzy prime ideal \( I_{\mu } \) in R have been constructed.

Prime uniserial modules and rings

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.

The Dual notion of fuzzy prime submodules

Let $R$ be a commutative ring and let $M$ be an $R$-module. In this paper, we introduce the dual notion of fuzzy prime (that is, fuzzy second) submodules of $M$ and explore some of the basic properties of this class of submodules. We say a non-zero fuzzy submodule $\mu$ of $M$ is fuzzy second if for each $r \in R$, we have $1_r.\mu = \mu$ or $1_r.\mu = 1_\theta$. It is shown that the fuzzy second submodules is a proper subclass of the fuzzy coprimary submodules.

Prime extension dimension of a module

We have that for a finitely generated module $M$ over a Noetherian ring $A$ any two RPE filtrations of $M$ have same length. We call this length as prime extension dimension of $M$ and denote it as $mr{pe.d}_A(M)$. This dimension measures how far a module is from torsion freeness. We show for every submodule (N) of (M), (mr{pe.d}_A(N)leqmr{pe.d}_A(M)) and (mr{pe.d}_A(N)+mr{pe.d}_A(M/N)geqmr{pe.d}_A(M)). We compute the prime extension dimension of a module using the prime extension dimensions of its primary submodules which occurs in a minimal primary decomposition of (0) in (M).

WE-Prime Submodules and WE-Semi-Prime Submodules

"In this article, "we introduce the concept of a WE-Prime submodule", as a stronger form of a weakly prime submodule". "And as a "generalization of WE-Prime submodule", we introduce the concept of WE-Semi-Prime submodule, which is also a stronger form of a weakly semi-prime submodule". "Various basic properties of these two concepts are discussed. Furthermore, the relationships between "WE-Prime submodules and weakly prime submodules" and studied". "On the other hand the relation between "WE-Prime submodules and WE-Semi-Prime submodules" are consider". "Also" the relation of "WE-Sime-Prime submodules and weakly semi-prime submodules" are explained. Behind that, some characterizations of these concepts are investigated".

On Graded Weakly 2-Absorbing Primary Submodules

In this paper, we introduce the concept of graded weakly 2-absorbing primary submodules as a generalization of graded weakly 2-absorbing primary ideals. A number of results concerning these classes of graded submodules and their homogeneous components are given.

On Nearly Prime Submodules of Unitary Modules

The aim of this paper is to introduce the notion of nearly prime submodules as a generalization of prime submodules. We investigate some of their basic properties and point out the similarities between these submodules and the prime submodules. We also indicate some applications of nearly prime submodules. These applications show how nearly prime submodules control the structure of modules and they recover earlier relative theorems.

On the graded classical prime spectrum of a graded module

Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper, we introduce and study a new topology on Cl.Specg(M), the collection of all graded classical prime submodules of M, called the Zariski-like topology. Then we investigate the relationship between algebraic properties of M and topological properties of Cl.Specg(M). Moreover, we study Cl.Specg(M) from point of view of spectral space.