H. Ansari-Toroghy, F. Farshadifar, S. Maleki-Roudposhti
Let R be a commutative ring with identity and M be an R-module. The main purpose of this paper is to introduce and investigate the notion of strongly ψ-2-absorbing second submodules of M as a generalization of strongly 2-absorbing second and ψ-second submodules of M.
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Let R be a commutative ring with identity, S be a multiplicatively closed subset of R, and let M be an R-module. In this paper, we introduce the notion of S-2-absorbing second submodules of M as a generalization of S-second submodules and strongly 2-absorbing second submodules of M. We investigate some properties of this class of submodules. Also, we obtain some results concerning S-2-absorbing submodules of M.
In this paper, we will introduce the concept of 2-absorbing (resp. strongly 2-absorbing) secondary submodules of modules over a commutative ring as a generalization of secondary modules and investigate some basic properties of these classes of modules.
Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary $R$-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule $N$ of $M$ is said to be a 1-absorbing primary submodule if whenever non-unit elements $a,b\in R$ and $m\in M$ with $abm\in N$, then either $ab\in(N:_{R}M)$ or $m\in M-rad(N).$ Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.
In this study, we aim to introduce the concept of a 1-absorbing prime submodule of an unital module over a commutative ring with a non-zero identity. Let M be an R-module and N be a proper submodule of M. For all non-unit elements a, b in R and m in M if abm in N, either ab in (N : M) or m in N, then N is called 1-absorbing prime submodule of M. We show that the new concept is a generalization of prime submodules at the same time it is a kind of special 2-absorbing submodule. In addition to some properties of a 1-absorbing prime submodule, we obtain a characterization of it in a multiplication…
All rings are commutative with $1\neq0$, and all modules are unital. The purpose of this paper is to investigate the concept of $2$-absorbing primary submodules generalizing $2$-absorbing primary ideals of rings. Let $M$ be an $R$-module. A proper submodule $N$ of an $R$-module $M$ is called a $2$-absorbing primary submodule of $M$ if whenever $a,b\in R$ and $m\in M$ and $abm\in N$, then $am\in M$-$rad(N)$ or $bm\in M$-$rad(N)$ or $ab\in(N:_RM)$. It is shown that a proper submodule $N$ of $M$ is a $2$-absorbing primary submodule if and only if whenever $I_1I_2K\subseteq N$ for some ideals $I_1…
Let $G$ be a group with identity $e$, $R$ be a commutative $G$-graded ring with unity $1$ and $M$ be a $G$-graded unital $R$-module. In this article, we introduce the concept of graded $1$-absorbing prime submodule. A proper graded $R$-submodule $N$ of $M$ is said to be a graded $1$-absorbing prime $R$-submodule of $M$ if for all non-unit homogeneous elements $x, y$ of $R$ and homogeneous element $m$ of $M$ with $xym\in N$, either $xy\in (N :_{R} M)$ or $m\in N$. We show that the new concept is a generalization of graded prime submodules at the same time it is a special graded $2$-absorbing su…
In this paper, we will introduce the concept of classical (resp. strongly classical) 2-absorbing second submodules of modules over a commutative ring as a generalization of 2-absorbing (resp. strongly 2-absorbing) second submodules and investigate some basic properties of these classes of modules.
The aim of this article is to introduce the concept of graded $2$-absorbing coprimary submodules as a generalization of graded strongly $2$-absorbing second submodules, and explore some properties of this class. A non-zero graded $R$-submodule $N$ of a graded $R$-module $M$ is called a graded $2$-absorbing coprimary $R$-submodule if whenever $x, y$ are homogeneous elements of $R$ and $K$ is a graded $R$-submodule of $M$ such that $xyN\subseteq K$, then either $x$ or $y$ is in the graded radical of $(K :_{R} N)$ or $xy\in Ann_{R}(N)$. Several results have been achieved.
Generalized square-difference factor absorbing submodules of modules over commutative rings
本文研究了交换环上模的广义平方差因子吸收子模的性质与行为，探讨了其在局部化、同态像、直积、理想化和融合等模扩展中的表现，并通过实例区分了该类子模与其他概念。
In this article, all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a classical prime submodule, if for each $m\in M$ and elements $a,b\in R$, $abm\in N$ implies that $am\in N$ or $bm\in N$. We introduce the concept of "classical 2-absorbing submodules" as a generalization of "classical prime submodules." We say that a proper submodule $N$ of $M$ is a classical 2-absorbing submodule if whenever $a,b,c\in R$ and $m\in M$ with $abcm\in N$, then $abm\in N$ or $acm\in N$ or $bcm\in N$.
In this work, we introduce the notion of $S$-1-absorbing primary submodule as an extension of 1-absorbing primary submodule. Let $S$ be a multiplicatively closed subset of a ring $R$ and $M$ be an $R$-module. A submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be $S$-1-absorbing primary if whenever $abm\in N$ for some non-unit $a,b\in R$ and $m\in M$, then either $sab\in(N:_{R}M)$ or $sm\in M$-$rad(N)$. We examine several properties of this concept and provide some characterizations. In addition, $S$-1-absorbing primary avoidance theorem and $S $-1-absorbing primary property for…
Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper we will obtain some results concerning the graded generalized 2-absorbing submodules and their homogeneous components. Special attention has been paid, when graded rings are graded gr-Noetherian, to find extra properties of these graded submodules.
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 classical and graded strongly classical 2-absorbing second submodules of graded modules over a graded commutative rings. A number of results concerning these classes of graded submodules and their homogeneous components are given.
On 2-absorbing submodules and their Amalgamations
本文研究了素子模推广的2-吸收子模，给出了其基本刻画，并利用同构与同态探讨了融合模中2-吸收子模的判定条件。
Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a graded unitary $R$-module. In this article, we introduce the concepts of graded $ϕ$-$2$-absorbing and graded $ϕ$-$2$-absorbing primary submodules as generalizations of the concepts of graded $2$-absorbing and graded $2$-absorbing primary submodules. Let $GS(M)$ be the set of all graded $R$-submodules of $M$ and $ϕ: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be a function. A proper graded $R$-submodule $K$ of $M$ is said to be a graded $ϕ$-$2$-absorbing $R$-submodule of $M$ if whenever $x, y$ are homogeneous elements of …
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring, $M$ a graded $R$-module and $A\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $A$-2-absorbing submodules of $M$ as a generalization of graded 2-absorbing submodules and graded $A$-prime submodules of $M.$ We investigate some properties of this class of graded submodules.
Let R be a commutative ring with unity and M be an R- module In this paper we introduce semi n- absorbing and (k, n)-closed submodules of modules over commutative rings, and investigate their basic properties.
In this paper, we introduce the concepts of strongly 2-absorbing primary ideals (resp., submodules) and strongly 2-absorbing ideals (resp., submodules) as generalizations of strongly prime ideals. Furthermore, we investigate some basic properties of these classes of ideals.
Some Characterizations of Weakly Pseudo Primary 2-Absorbing Submodules in Terms of some Types of Modules
本文研究弱伪准素2-吸收子模，利用忠实、非奇异、Z-正则及投射模刻画乘法模，并在附加条件下证明其子模剩余为相应理想。
Archimedean classes in additive monoids
本文针对加法幂等半环上模中常见的被和吸收子模，推进了对该类子模及更广泛加法幺半群的研究，重点探讨其阿基米德类与商结构。
Let R be a commutative ring with identity and M be an R-module. In this paper, we will introduce the concept of 2-irreducible (resp., strongly 2- irreducible) submodules of M as a generalization of irreducible (resp., strongly irreducible) submodules of M and investigated some properties of these classes of modules.
Let $R$ be a commutative ring with identity. For an $R$-module $M$, the notion of strongly prime submodule of $M$ is defined. It is shown that this notion of prime submodule inherits most of the essential properties of the usual notion of prime ideal. In particular, the Generalized Principal Ideal Theorem is extended to modules.
The purpose of the present paper is to prove some properties of the strongly irreducible submodules in the arithmetical and Noetherian modules over a commutative ring. The relationship among the families of strongly irreducible submodules, irreducible submodules, prime submodules and primal submodules is proved. Also, several new characterizations of the arithmetical modules are given. In the case when $R$ is Noetherian and $M$ is finitely generated, several characterizations of strongly irreducible submodules are included. Among other things, it is shown that when $N$ is a submodule of $M$ su…
This article aims to solve positively Anderson-Badawi Conjecture of n-Absorbing and strongly n-absorbing ideals of commutative rings in the class of u-rings. The main result extends and recovers Anderson-Badawis related result on Prufer domains
Let $R$ be a commutative ring and $M$ a non-zero $R$-module. We introduce the class of \emph{pseudo strongly hollow submodules} (\emph{PS-hollow submodules}, for short) of $M$. Inspired by the theory of modules with \emph{secondary representations}, we investigate modules which can be written as \emph{finite} sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of \emph{minimal} PS-hollow strongly representations of modules over Artinian rings.
Collections of an Ideal: Any $n$-Absorbing Ideal is Strongly $n$-Absorbing
该研究引入极大与半极大集合工具，结合前人成果证明了Anderson和Badawi关于吸收理想的首个猜想——任意$n$-吸收理想都是强$n$-吸收的，并提出三个蕴含其最后一个未解决猜想的新猜想。
Let $G$ be a group with identity $e$ and $R$ be a $G$-graded commutative ring with nonzero unity $1$. In this article, we introduce the concept of graded strongly $1$-absorbing primary ideals. A proper graded ideal $P$ of $R$ is said to be a graded strongly $1$-absorbing primary ideal of $R$ if whenever nonunit homogeneous elements $x, y, z\in R$ such that $xyz\in P$, then either $xy\in P$ or $z\in Grad(\{0\})$ (the graded radical of $\{0\}$). Several properties of graded strongly $1$-absorbing primary ideals are investigated. Many results are given to disclose the relations between this new c…
We introduce the notion of a (strongly) topological lattice $\mathcal{L}=(L,\wedge ,\vee)$ with respect to a subset $X\subsetneqq L;$ aprototype is the lattice of (two-sided) ideals of a ring $R,$ which is(strongly) topological with respect to the prime spectrum of $R.$ We investigate and characterize (strongly) topological lattices. Given a non-zero left $R$-module $M,$ we introduce and investigate the spectrum $\mathrm{Spec}^{\mathrm{f}}(M)$ of \textit{first submodules} of $M.$ We topologize $\mathrm{Spec}^{\mathrm{f}}(M)$ and investigate the algebraic properties of $_{R}M$ by passing to the…
Let R be a multiplicative hyperring. In this paper, we define the concept of 1-absorbing prime hyperideals which is a generalization of the prime hyperideals. Several properties of the hyperideals are provided. Moreover, we introduce the notions of 1-absorbing primary hyperideals, strongly 1-absorbing primary hyperideals and weakly 1-absorbing primary hyperideals which are generalizations of the 1-absorbing prime hyperideals and then we will show some properties of them.
Commutative rings with square-difference factor absorbing factorization
该研究引入SAF-环与SSAF-环，明确其与SP-环等经典环类的联系，证明诺特SSAF-环等价于ZPI-环，并给出相关应用。
Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings
本文在强离散凝聚环上推广Buchberger算法，证明首项模可数生成并给出计算生成集的算法，同时给出Hilbert合冲定理的构造性版本。
Strongly multiplicative sets, idempotent localizations, and $S$-prime phenomena
本文研究交换环的强乘法集，刻画其与幂等局部化及$S$-素理想的关系，并给出保持性结果、挠理论及Krull分离引理等应用。
Characterizations of t-Almost Dedekind Domains via w-Secondary Modules
该研究通过w-二次模的性质，刻画了t-几乎戴德金整环的等价条件，明确了w-可表示模的局部化与强可表示性的关联。
On Golomb Topology of Modules over Commutative Rings
该研究为交换环上的非零幺模定义了Golomb拓扑，证明了特定条件下互素陪集族可构成拓扑基，同时探究了该拓扑性质与模的代数性质的关联，并用以刻画几类重要模。
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