جستجوی مقالات مرتبط با کلیدواژه "‎multiplication module" در نشریات گروه "ریاضی"

‎Let $R$ be a commutative ring‎, ‎$M$ an $R$-module‎, ‎and $n\geq 1$ an integer‎. ‎In this paper‎, ‎we will introduce the concept of $n$-pure submodules of $M$ as a generalization of pure submodules and obtain some related results‎.‎We say that a submodule $N$ of $M$ is a \emph {$n$-pure submodule of $M$} if $I_1I_2...I_nN=I_1N \cap I_2N\cap...I_nN\cap (I_1I_2...I_n)M$ for all proper ideals $I_1‎, ‎I_2,...I_n$ of $R$‎.

‎In this paper‎, ‎our aim is to introduce and study the essential submodules of an $R$-module $M$ relative to an arbitrary submodule $T$ of $M$‎. ‎Let $T$ be an arbitrary submodule of an $R$-module $M$‎, ‎then we say that a submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎whenever for every submodule $X$ of $M$‎, ‎$N\cap X\subseteq T$ implies that‎ ‎$(T:M)\subseteq ^{e}{\rm Ann}(X)$‎. ‎We investigate some new results concerning to this class of submodules‎. ‎Among various results we prove that for a faithful multiplication $R$-module $M$‎, ‎if the submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎then $(N:M)$ is an essential ideal of $R$ relative to $(T:M)$‎. ‎The converse is true if $M$ is moreover a finitely generated module‎.

‎‎Let $R$ be a commutative ring with non-zero identity, $S\subseteq R$ be a multiplicatively closed subset of $R$ and let $M$ be an $R$-module.  A submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be almost $S$-prime, if there exists an $s\in S$ such that whenever $rm\in N-(N:_{R}M)N$, then $sm\in N$ or $sr\in (N:_{R}M)$ for each $r\in R$, $m\in M$. The aim of this paper is to introduce and investigate some properties of the notion of almost $S$-prime submodules, especially in multiplication modules. Moreover, we investigate the behaviour of this structure under module homomorphisms, localizations, quotient modules, Cartesian product. Finally, we state two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are almost $S$-prime.

Let R be a commutative ring with identity, and let W be a unitary R-module. In this paper, we introduced the concept of a weakly semi-primary submodule as a generalization of the primary submodule, where a submodule X of W is called weakly semi-primary if the Rad(X:W)=(X:W)−−−−−−−√ is a weakly prime ideal of R, and from this work, we have provided some characteristics of weakly semi-primary submodule.

In this paper, we introduce the concept of primary submodules over S which is a generalization of the concept of S-prime submodules. Suppose S is a multiplicatively closed subset of a commutative ring R and let M be a unital R-module. A proper submodule Q of M with (Q :R M) ∩ S = ∅ is called primary over S if there is an s ∈ S such that, for all a ∈ R, m ∈ M, am ∈ Q implies that sm ∈ Q or san ∈ (Q :R M), for some positive integer n. We get some new results on primary submodules over S. Furtheremore, we compare the concept of primary submodules over S with primary ones. In particular, we show that a submodule Q is primary over S if and only if (Q :M s) is primary, for some s ∈ S.

Let R be a commutative ring with identity and n a positive integer greater than 1. In this paper, we introduce the concept of semi-n-absorbing submodules. A proper submodule N of an R-module M is called a semi-n-absorbing submodule of M if whenever a \in R, x \in M and a^nx \in N then ax \in N ora^n \in (N :R M). A number of results concerning semi-n-absorbing submodulesand examples of them are given. It is shown that if N is a semi-n-absorbing submodule of M and F is a flat R-module such that F \otimes N is proper in F \otimes M then F \otimes N is semi-n-absorbing in F \otimes M. We show that the converse holds when F is faithfully flat.

Let 𝑅 be a commutative ring with identity and 𝑀 be a unitary 𝑅-module,and let 𝐼(𝑅)* be the set of all nontrivial ideals of 𝑅. The complement ofthe 𝑀-intersection graph of ideals of 𝑅, denoted by Γ𝑀(𝑅), is a graphwith the vertex set 𝐼(𝑅)* , and two distinct vertices 𝐼 and 𝐽 are adjacentif and only if 𝐼𝑀∩𝐽𝑀={0}. In this paper, for every multiplication𝑅-module 𝑀, the diameter and the girth of Γ𝑀(𝑅) are determined. Also,we show that if 𝑚,𝑛>1 are two integers and ℤ𝑛 is a ℤ𝑚-module, thenthe complement of the ℤ𝑛-intersection graph of ideals of ℤ𝑚 is weakly perfect.

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N = IM$. It is shown that over a Noetherian domain $R$ with dim$(R)leq 1$, multiplication modules are precisely cyclic or isomorphic to an invertible ideal of $R$. Moreover, we give a characterization of finitely generated multiplication modules.

Let R be a commutative ring with identity and M be a unitary R-module. In this paper we generalize the concept multiplicatively closed subset of R and we study some properties of these genaralized subsets of M. Among the many results in this paper, we generalize some well-known theorems about multiplicatively closed subsets of R to these generalized subsets of M. Also we show that some other well-known results about multiplicatively closed subsets of R are not valid for these generalized subsets of M.

Let R be a commutative ring with identity and M be a finitely generated unital R-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module M is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals of modules which are the direct sum of finitely generated module and finitely generated multiplication module are shown. Finally, we study some properties of modules that are the direct sum of multiplication modules in terms of Fitting ideals.

Let R be a commutative ring and let M be an R-module. We define the small intersection graph G(M) of M with all non-small proper submodules of M as vertices and two distinct vertices N,K are adjacent if and only if N∩K is a non-small submodule of M. In this article, we investigate the interplay between the graph-theoretic properties of G(M) and algebraic properties of M, where Mis a multiplication module.

We state several conditions under which comultiplication and weak comultiplication modules are cyclic and study strong comultiplication modules and comultiplication rings. In particular, we will show that every faithful weak comultiplication module having a maximal submodule over a reduced ring with a finite indecomposable decomposition is cyclic. Also we show that if M is an strong comultiplication R-module, then R is semilocal and M is finitely cogenerated. Furthermore, we define an R-module M to be p-comultiplication, if every nontrivial submodule of M is the annihilator of some prime ideal of R containing the annihilator of M and give a characterization of all cyclic p-comultiplication modules.
Moreover, we prove that every pcomultiplication module which is not cyclic, has no maximal submodule and its annihilator is not prime. Also we give an example of a module over a Dedekind domain which is not weak comultiplication, but all of whose localizations at prime ideals are comultiplication and hence serves as a counterexample to [10, Proposition 2.3] and [11, Proposition 2.4].

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. The primary-like spectrum $Spec_L(M)$ is the collection of all primary-like submodules $Q$ such that $M/Q$ is a primeful $R$-module. Here, $M$ is defined to be RSP if $rad(Q)$ is a prime submodule for all $Qin Spec_L(M)$. This class contains the family of multiplication modules properly. The purpose of this paper is to introduce and investigate a new Zariski space of an RSP module, called Zariski-like space. In particular, we provide conditions under which the Zariski-like space of a multiplication module has a subtractive basis.

Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly. In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful property.

Let R be a commutative ring with identity and M be a unitary R-module. A proper submodule N of M is 2− absorbing if r 1, r2, r3 2 R, m 2 M with r1r2r3m 2 M implies r1r2m 2 N or r1r3m 2 N or r2r3m 2 N. Let ' : S(M) −! S(M) [ {ø} be a function where S (M) is the set of all submodules of M. We call a proper submodule N of M a '-2-absorbing submodule if r1, r2, r3 2 R, m 2 M with r 1r2r3m 2 N −'(N) implies that r1r2m 2 N or r1r3m 2 N or r2r3m 2N. We want to extend 2-absorbing ideals to '-2-absorbing submodules and we show that-2-absorbing submodules enjoy analogs of many ofthe properties of 2-absorbing ideals.