Prime Submodules Research Articles

Graded pseudo weakly prime spectrum of graded topological modules

In this study, we introduce graded pseudo weakly prime submodules of G-graded R-modules, which are an extension of graded weakly prime ideals over G-graded rings. On the graded spectrum of graded pseudo weakly prime submodules, we investigate the Zariski topology. Different aspects of this topological space are investigated, and they are linked to the algebraic properties of the G-graded R-modules under study.

Zariski topology on the spectrum of intuitionistic fuzzy classical primary submodules

Zariski topology on the spectrum of intuitionistic fuzzy classical primary submodules

A scheme associated to modules over commutative rings

For any module M over a commutative ring R with identity, we consider Sch ( M ) as a certain class of prime submodules modulo the equivalence relation that two such prime submodules are the same if their colon ideals are equal. We topologize Sch ( M ) and introduce a sheaf O Sch ( M ) of commutative rings on it, which makes ( Sch ( M ) , O Sch ( M ) ) into a scheme. In particular, if M is a module over a Noetherian ring R, then ( Sch ( M ) , O Sch ( M ) ) is a locally Noetherian scheme. Among others, we give sufficient conditions for Sch ( M ) to be an affine scheme.

Prime submodules and maximal submodules in the idealization of a module

This paper describes certain types of prime submodules and maximal submodules that are found within the idealization of a module. As a consequence of this, it is concluded that every module over a commutative ring with identity element has an extension whose prime spectrum is nonempty.

Classical 1-Absorbing Primary Submodules

Over the years, prime submodules and their generalizations have played a pivotal role in commutative algebra, garnering considerable attention from numerous researchers and scholars in the field. This papers presents a generalization of 1-absorbing primary ideals, namely the classical 1-absorbing primary submodules. Let ℜ be a commutative ring and M an ℜ-module. A proper submodule K of M is called a classical 1-absorbing primary submodule of M, if xyzη∈K for some η∈M and nonunits x,y,z∈ℜ, then xyη∈K or ztη∈K for some t≥1. In addition to providing various characterizations of classical 1-absorbing primary submodules, we examine relationships between classical 1-absorbing primary submodules and 1-absorbing primary submodules. We also explore the properties of classical 1-absorbing primary submodules under homomorphism in factor modules, the localization modules and Cartesian product of modules. Finally, we investigate this class of submodules in amalgamated duplication of modules.

The Characterization of Almost Prime Submodule on the 6 Finitely Generated Module over Principal Ideal Domain

The Characterization of Almost Prime Submodule on the 6 Finitely Generated Module over Principal Ideal Domain

Rad-Quasi-Prime Submodules

Consider a left J-module I. The present study introduces the conception of rad-Quasi- Prime submodule, that serves as a dual popularization of both Quasi-Prime submodules and primary submodules. An apposite submodule A of an J-module named as rad- Quasi Prime if for all and with implies that either or . Numerous facts and characterizations that concerning are acquired.

Some remarks on S-strongly prime submodules

Some remarks on S-strongly prime submodules

Minimal Prime and Semiprime Submodules

Prime and semiprime submodules are important generalizations of prime and semiprime ideals to module theory over commutative rings. However, minimal or smallest prime/semiprime submodules have received comparatively less attention. This paper furthers the theory of minimal prime and minimal semiprime submodules through several new directions. After formally introducing these concepts, we establish characterization theorems for minimal prime submodules based on intersections of primes and associated primes. A complete structure theory is also developed for minimal semiprime submodules, allowing their description up to isomorphism. We introduce prime and semiprime operators which stably preserve primeness properties between modules. Results on the persistence of minimal primeness of submodules under such operators are proved. Connections of our minimal semiprime characterization back to classic ring theory are highlighted, recovering past results on minimal prime ideals. Overall, this paper significantly expands the foundations of minimal prime and semiprime submodules, while opening up many new questions at this nexus of module and ring theory.

Some remarks on the dual notion of n-absorbing primary submodules

Let [Formula: see text] be a positive integer. In this paper, we introduce a functional method, which gives us an equivalent condition to the concept of [Formula: see text]-absorbing primary submodules. By using this method, the dual notion of [Formula: see text]-absorbing primary submodules, which we call [Formula: see text]-secondary submodules, is presented. Then, some properties of this class of submodules, several useful examples and functorial results are presented.

On intuitionistic fuzzy almost prime ideals and intuitionistic fuzzy almost prime submodules

The aim of this paper is to present some characterizations of almost prime ideals and almost prime submodules in the intuitionistic fuzzy environment. We investigate various properties of these concepts and achieve many results.

A Note on Minimal Prime Submodules

In this short note, we give a characterization of the modules with finitely many minimal prime submodules over an arbitrary submodule.

1-absorbing and weakly 1-absorbing prime submodules of a module over a noncommutative ring

In this study, we aim to introduce the concepts of 1-absorbing prime submodules and weakly 1-absorbing prime submodules of a unital module over a noncommutative ring with nonzero identity. This is a new class of submodules between prime submodules (weakly prime submodules) and 2-absorbing submodules (weakly 2-absorbing submodules). Let R be a noncommutative ring with a nonzero identity 1ne 0 and M an R-module. A proper submdule P of M is said to be a 1-absorbing prime submodule (weakly 1-absorbing prime submodule) if for all nonunits x,yin R and min M with xRyRmsubseteq P(left{ 0right} ne xRyRmsubseteq P), then xyin (M:_{R}P) or min P. Various properties and characterizations of these classes of submodules are considered.

Intuitionistic L-fuzzy classical prime and intuitionistic L-fuzzy 2-absorbing submodules

Let L be a complete lattice. We introduce and characterise intuitionistic L-fuzzy classical prime submodule and intuitionistic L-fuzzy 2-absorbing submodules of a unitary module M over a commutative ring R with identity. We compare both of these submodules with intuitionistic L-fuzzy prime submodules. It is proven that in the case of the multiplication module M, the two notions of intuitionistic L-fuzzy classical prime submodules and intuitionistic L-fuzzy prime submodules coincide. Many other related results concerning these notions are obtained.

On gw-Prime Submodules

Our aim in this work is to investigate prime submodules and prove some properties of them. We study the relations between prime submodules of a given module and the extension of prime submodules. The relations between prime submodules of two given modules and the prime submodules in the direct product of their quotient module are studied and investigated.

2-Prime Modules

In this paper, we introduce the notion of a 2-prime module as a generalization of prime module E over a ring R, where E is said to be prime module if (0) is a prime submodule. We introduced the concept of the 2-prime R-module. Module E is said to be 2-prime if (0) is 2-prime submodule of E. where a proper submodule K of module E is 2-prime submodule if, whenever rR, xE, E, Thus xK or [K: E].

δ-small submodule and prime modules

In this paper, we introduced and studied δ-small submodule over prime module. Two concepts are very important namely strongly prime submodule and completely prime submodule. Multiple results led to obtaining a δ-small submodule of a singular, divisible and Bezout module with R is local. Important terms that appeared in this article, together with some terms, produced the submodule that we were interested in.

Strong Essential Submodules And Strong Uniform Modules

A non-zero submodule K of an R-module M is called essential if K L (0) for each non-zero submodule L of M . And an R-module M is called uniform if each non-zero submodule of M is an essential . In this paper we give generalization of essential submodule and uniform module that are strong essential submodule and strong uniform module. A non-zero submodule N of M is called strong essential if N P (0) for each non-zero strongly prime submodule P of M . And an R-module M is called strong uniform if each non-zero submodule of M is a strong essential .

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".

Nearly Quasi 2-Absorbing submodule

All rings in this note are commutative rings with identity, and all Rmodules are left unitary. "A proper submodule E of an R-module X iscalled nearly quasi prime submodule, if whenever abx ∈ E, with a, b ∈