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 $m_{i}\in h(M).$

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 S-1-absorbing prime submodules

Let [Formula: see text] be a commutative ring with non-zero identity, [Formula: see text] a multiplicatively closed subset of [Formula: see text] and [Formula: see text] a unital [Formula: see text]-module. In this paper, we introduce and study the concept of [Formula: see text]-1-absorbing prime submodules. A submodule [Formula: see text] of [Formula: see text] with [Formula: see text] is said to be [Formula: see text]-1-absorbing prime, if there exists an [Formula: see text] whenever [Formula: see text] for some non-unit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text]. Some examples, characterizations and properties of [Formula: see text]-1-absorbing prime submodules are given. Moreover, we give some characterizations of [Formula: see text]-1-absorbing prime submodules in multiplication modules.

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.

Zariski topology on the spectrum of graded pseudo prime submodules

In this article, we introduce the concept of graded pseudo prime submodules of graded modules that is a generalization of the graded prime ideals over commutative rings. We study the Zariski topology on the graded spectrum of graded pseudo prime submodules. We clarify the relation between the properties of this topological space and the algebraic properties of the graded modules under consideration.

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>