On P-Essential Submodules

Let be a commutative ring with identity and let be an R-module. We call an R-submodule of as P-essential if for each nonzero prime submodule of and 0 . Also, we call an R-module as P-uniform if every non-zero submodule of is P-essential. We give some properties of P-essential and introduce many properties to P-uniform R-module. Also, we give conditions under which a submodule of a multiplication R-module becomes P-essential. Moreover, various properties of P-essential submodules are considered.

ON GRADED S-PRIME SUBMODULES OF GRADED MODULES OVER GRADED COMMUTATIVE RINGS

Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity, $M$ a graded $R$-module and $S\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $S$-prime submodules of graded modules over graded commutative rings. We investigate some properties of this class of graded submodules and their homogeneous components. Let $N$ be a graded submodule of $M$ such that $(N:_{R}M)\cap S=\emptyset $. We say that $N$ is \textit{a graded }$S$\textit{-prime submodule of }$M$ if there exists $s_{g}\in S$ and whenever $a_{h}m_{i}\in N,$ then either $s_{g}a_{h}\in (N:_{R}M)$ or $s_{g}m_{i}\in N$ for each $a_{h}\in h(R) $ and $m_{i}\in h(M).$

A note on almost prime submodule of CSM module over principal ideal domain

An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

On \phi prime submodules

In this paper, we study some properties of φ-prime submodules andwe give another charactrization for it. For given submodules N and K of a moduleM with K ⊆ N, we prove that N is φ-prime submodule if and only if N/Kis φ_K-prime submodule. Finally, we show that any finite sum of φ-prime submodules isφ-prime.

SOME REMARKS ON THE CLASSICAL PRIME SPECTRUM OF MODULES

Let R be a commutative ring with identity and let M be an R-module. A proper submodule P of M is called a classical prime submodule if abm ∈ P, for a,b ∈ R, and m ∈ M, implies that am ∈ P or bm ∈ P. The classical prime spectrum of M, Cl.Spec(M), is defined to be the set of all classical prime submodules of M. We say M is classical primefule if M = 0, or the map ψ from Cl.Spec(M) to Spec(R/Ann(M)), defined by ψ(P) = (P : M)/Ann(M) for all P ∈ Cl.Spec(M), is surjective. In this paper, we study classical primeful modules as a generalisation of primeful modules. Also we investigate some properties of a topology that is defined on Cl.Spec(M), named the Zariski topology.

On graded Ie-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 with identity and M a graded R-module. We introduce the concept of graded Ie-prime submodule as a generalization of a graded prime submodule for I =?g?G Ig a fixed graded ideal of R. We give a number of results concerning this class of graded submodules and their homogeneous components. A proper graded submodule N of M is said to be a graded Ie-prime submodule of M if whenever rg ? h(R) and mh ? h(M) with rgmh ? N ? IeN, then either rg ? (N :R M) or mh ? N.

On $ \Phi $-powerful submodules and $ \mathrm{\Phi} $-strongly prime submodules

<abstract><p>Let $ R $ be a commutative ring with identity and $ N $ be a submodule of an $ R $-module $ M $. We say a nonnil submodule $ N $ of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-powerful (resp., $ \mathrm{\Phi} $-strongly prime) submodule, if $ \mathrm{\Phi}(N) $ is a powerful (resp., strongly prime) submodule of a module $ \mathrm{\Phi}(M) $. We show that a nonnil prime submodule $ N $ of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-powerful submodule if and only if it is a $ \mathrm{\Phi} $-strongly prime submodule. Similarly, if every prime submodule of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-strongly prime, then we call it a $ \mathrm{\Phi} $-pseudo-valuation module ($ \mathrm{\Phi} $-PVM). We also prove that a faithful multiplication $ R $-module $ M $ is $ \mathrm{\Phi} $-PVM if and only if some maximal nonnil submodules of $ M $ are $ \mathrm{\Phi} $-powerful. In this perspective, we analyze that $ M $ is $ \mathrm{\Phi} $-PVM if and only if $ R $ is a PVD. In due course, we provide some characterizations of these submodules along with their relationships under certain conditions.</p></abstract>

On the prime spectrum of an le-module

Here, we continue to characterize a recently introduced notion, le-modules [Formula: see text] over a commutative ring [Formula: see text] with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga. 158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec[Formula: see text] of all prime submodule elements of [Formula: see text]. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec[Formula: see text] is connected if and only if [Formula: see text] contains no idempotents other than [Formula: see text] and [Formula: see text]. Open sets in the Zariski topology for the quotient ring [Formula: see text] induces a base of quasi-compact open sets for the Zariski topology on Spec[Formula: see text]. Every irreducible closed subset of Spec[Formula: see text] has a generic point. Besides, we prove a number of different equivalent characterizations for Spec[Formula: see text] to be spectral.

Sifat-sifat Submodul Prima dan Submodul Prima Lemah

Penelitian ini merupakan penelitian kajian pustaka yang bertujuan untuk mengkaji sifat-sifat submodul prima dan submodul prima lemah serta hubungan antara keduanya. Kajian dimulai dari definisi submodul prima dan submodul prima lemah, selanjutnya dikaji mengenai sifat-sifat dari keduanya. Pada penelitian ini, semua ring yang diberikan adalah ring komutatif dengan unsur kesatuan dan modul yang diberikan adalah modul uniter. Sebagai hasil dari penelitian ini diperoleh beberapa pernyataan yang ekuivalen, misalkan suatu -modul , submodul sejati di dan ideal di , maka ketiga pernyataan berikut ekuivalen, (1) merupakan submodul prima, (2) Setiap submodul tak nol dari -modul memiliki annihilator yang sama, (3) Untuk setiap submodul di , subring di , jika berlaku maka atau . Di lain hal, pada submodul prima lemah jika diberikan suatu -modul, submodul sejati di , maka pernyataan berikut ekuivalen, yaitu (1) Submodul merupakan submodul prima lemah, (2) Untuk setiap , jika maka . Selain itu, didapatkan pula hubungan antara keduanya, yaitu setiap submodul prima merupakan submodul prima lemah.Kata Kunci: Submodul Prima, Submodul Prima Lemah, Ideal Prima. This research is literature study that aims to examine the properties of prime submodules and weakly prime submodules and the relationship between both of them. The study starts from the definition of prime submodules and weakly prime submodules, then reviewed about the properties both of them. Throughout this paper all rings are commutative with identity and all modules are unitary. As the result of this research, obtained several equivalent statements, let be a -module, be a proper submodule of and ideal of , then the following three statetments are equivalent, (1) is a prime submodule, (2) Every nonzero submodule of -module has the same annihilator, (3) For any submodule of , subring of , if then or . In other case, for weakly prime submodules, if given is a unitary -module, be a proper submodule of , then the following statements are equivalent, (1) is a weakly prime submodule, (2) For any , if then . In addition, also found the relationship between both of them, i.e. any prime submodule is weakly prime submodule.Keywords: Prime Submodules, Weakly Prime Submdules, Prime Ideal.