Weakly Q-ideals of commutative rings
Mahdi Anbarloei
引入交换环中弱Q-理想的概念，研究其基本性质及与Q-理想的关系。
本文引入并研究了交换环中弱Q-理想的概念。
In this paper, we introduce and study the notion of weakly Q-ideals in commutative rings.
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All rings are commutative with $1\neq0$. The purpose of this paper is to investigate the concept of weakly $n$-absorbing ideals generalizing weakly 2-absorbing ideals. We prove that over a $u$-ring $R$ the Anderson-Badawi's conjectures about $n$-absorbing ideals and the Badawi-Yousefian's question about weakly 2-absorbing ideals hold.
In this study, we introduce the concept of "uniformly 2-absorbing primary ideals" of commutative rings, which imposes a certain boundedness condition on the usual notion of 2-absorbing primary ideals of commutative rings. Then we investigate some properties of uniformly 2-absorbing primary ideals of commutative rings with examples. Also, we investigate a specific kind of uniformly 2-absorbing primary ideals by the name of "special 2-absorbing primary ideals".
(n,Q)-ideals ans phi-(n,Q)-ideals of commutative rings
引入并研究交换环中(n,Q)-理想与phi-(n,Q)-理想的概念，探讨其基本性质及与经典理想的关系。
A famous result due to I. M. Isaacs states that if a commutative ring $R$ has the property that every prime ideal is principal, then every ideal of $R$ is principal. This motivates ring theorists to study commutative rings for which every ideal is a direct sum of cyclically presented modules. In this paper, we study commutative rings whose ideals are direct sum of cyclically presented modules.
Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. By $J(R),$ we denote the Jacobson radical of $R$. The purpose of this paper is to introduce the concept of weakly $J$-submodules generalizing $J$-submodules. We call a proper submodule $N$ of $M$ a weakly $J$-submodule if whenever $0\neq rm\in N$ for $r\in R$ and $m\in M$, then $r\in(J(R)M:M)$ or $m\in N.$ Various properties and characterizations of weakly $J$-submodules are investigated especially in the case of multiplication modules.
In commutative ring theory, there is a theorem of Cohen which states that if in a commutative ring all prime ideals are finitely generated then every ideal is finitely generated. However, it is known that having only maximal ideals finitely generated doesn't imply all ideals be finitely generated. In his article we ask the question that what happens if we assume all non-maximal ideals are finitely generated, and we answer the question by showing that indeed then all maximal ideals are also finitely generated i.e. the ring becomes Noetherian.
Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this paper, we introduce the cozero-divisor graph $\acuteΓ_I(R)$ of $R$ and obtain some related results.
Let $R$ be a commutative ring with non-zero identity. In this paper, we introduce the concept of weakly $J$-ideals as a new generalization of $J$-ideals. We call a proper ideal $I$ of a ring $R$ a weakly $J$-ideal if whenever $a,b\in R$ with $0\neq ab\in I$ and $a\notin J(R)$, then $a\in I$. Many of the basic properties and characterizations of this concept are studied. We investigate weakly $J$-ideals under various contexts of constructions such as direct products, localizations, homomorphic images. Moreover, a number of examples and results on weakly $J$-ideals are discussed. Finally, the th…
Cubes-difference factor absorbing ideals of a commutative ring
本文在交换环理论中引入并研究了立方差因子吸收理想，将其作为素理想概念的推广。
Let R be a commutative ring with identity. In this paper, we introduce the concept 1-absrbing primary ideal of R.
Second ideal intersection graph of a commutative ring
本文定义了交换环的第二理想交图，其顶点为非零真理想，边由交为第二理想决定，并研究了其基本性质。
The main purpose of this paper is to investigate prime, primary, and maximal ideals of semirings. The localization and primary decomposition of ideals in semirings are also studied.
For commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if whenever $a^{2}\in I$ and $Ann_{R}(a)=0$, then $a\in I$. Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi $r$-ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi $r$-ideals of rings to …
We define a new generalization of n-absorbing ideals in commutative rings called n-absorbing I-primary ideals. We investigate some characterizations and properties of such new generalization. If P is an n-absorbing I-primary ideal of R and $\sqrt{IP} =I \sqrt{P} $, then $\sqrt{P}$ is a n-absorbing I-primary ideal of R. Also, if $\sqrt{P}$ is an (n-1)-absorbing ideal of R such that $\sqrt{I \sqrt{P}} \subseteq IP$, then P is an n-absorbing I-primary ideal of R.
In this study, we present the generalization of the concept of $r$-ideals in commutative rings with nonzero identity. Let $R$ be a commutative ring with $0\neq1$ and $L(R)$ be the lattice of all ideals of $R$. Suppose that $ϕ:L(R)\rightarrow L(R)\cup\left\{\emptyset\right\}$ is a function. A proper ideal $I$ of $R$ is called a $ϕ-r$-ideal of $R$ if whenever $ab\in I$ and $Ann(a)=(0)$ imply that $b\in I$ for each $a,b\in R.$ In addition to giving many properties of $ϕ-r$-ideal, we also examine the concept of $ϕ-r$-ideal in trivial ring extension and use them to characterize total quotient rings…
Let $R$ be a commutative ring with nonzero identity, and $δ:\mathcal{I(R)}\rightarrow\mathcal{I(R)}$ be an ideal expansion where $\mathcal{I(R)}$ the set of all ideals of $R$. In this paper, we introduce the concept of $δ$-$n$-ideals which is an extension of $n$-ideals in commutative rings. We call a proper ideal $I$ of $R$ a $δ$-$n$-ideal if whenever $a,b\in R$ with$\ ab\in I$ and $a\notin\sqrt{0}$, then $b\in δ(I)$. For example, $δ_{1}$ is defined by $δ_{1}(I)=\sqrt{I}.$ A number of results and characterizations related to $δ$-$n$-ideals are given. Furthermore, we present some results relate…
(Weakly) $(α,β)$-prime hyperideals in commutative multiplicative hypeering
本文在交换乘法超环中引入（弱）$(α,β)$-素超理想的概念，研究了其基本性质，扩展了超环超理想的理论框架。
We propose a novel approach to distinguish table vs non-table ideals by using different machine learning algorithms. We introduce the reader to table ideals, assuming some knowledge on commutative algebra and describe their main properties. We create a data set containing table and non-table ideals, and we use a feedforward neural network model, a decision tree and a graph neural networks for the classification. Our results indicate that there exists an algorithm to distinguish table ideals from non-table ideals, and we prove it along some novel results on table ideals.
In this paper we study the concept of radical factorization in the context of abstract ideal theory in order to obtain a unified approach to the theory of factorization into radical ideals and elements in the literature of commutative rings, monoids and ideal systems. Using this approach we derive new characterizations of classes of rings whose ideals are a product of radical ideals, and we obtain also similar characterizations for classes of ideal systems in monoids and star ideals in integral domains.
Quasi S-n-ideals in commutative semirings
本文在带单位元的交换半环中引入拟S-n-理想概念，研究其与相关理想的关系、局部化性态及商多项式半环中的情况，还探讨其通过理想化与融合构造的自然扩张。
Let S be a set of n ideals of a commutative ring A. Let G_{even} (respectively G_{odd}) denote the product of all the sums of even (respectively odd) number of ideals of S. If n<7 the product of G_{even} and the intersection of all ideals of S is included in G_{odd}. In the case A is an Noetherian integral domain, this inclusion is replaced by equality if and only if A is a Dedekind domain.
Let $R$ be a commutative ring with $1\neq0$. In this article, we introduce the concept of weakly $(m,n)-$closed $δ-$primary ideals of $R$ and explore its basic properties. We show that $I\bowtie^{f}J$ is a weakly $(m,n)-$closed $δ_{\bowtie^{f}}-$primary ideal of $A\bowtie^{f}J$ that is not $(m,n)-$closed $δ_{\bowtie^{f}}-$primary if and only if $I$ is a weakly $(m,n)-$closed $δ-$primary ideal of $A$ that is not $(m,n)-$closed $δ-$primary and for every $δ$-$(m,n)$-unbreakable-zero element $a$ of $I$ we have $(f(a)+j)^{m}=0$ for every $j\in~J$, where $f:A\rightarrow B$ is a homomorphism of rings…
Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Under various hypotheses, it is proved that the center of $\mbox{End}_R(M)$ coincides with the endomorphism ring of the trace ideal of $M$. These results are exploited to establish results for balanced and rigid modules, and to settle certain cases of a conjecture of Huneke and Wiegand.
Let $R$ be a commutative ring with identity. An ideal $I$ of $R$ is said to be a big ideal (resp. an upper big ideal) if whenever $J\subsetneqq I$ (resp. $I\subsetneqq J$), $J^{n}\subsetneqq I^{n}$ (resp. $I^{n}\subsetneqq J^{n}$) for every $n\geq 1$; and $R$ itself is a big ideal ring provided that every ideal of $R$ is a big ideal. In this paper we study the notions of big ideals, upper big ideals and big ideal rings in different contexts of commutative rings such us integrally closed domains, pullbacks and trivial ring extensions etc. We show that the notions of big and upper big ideals are…
Let $R$ be a commutative chain ring. We use a variation of Gröbner bases to study the lattice of ideals of $R[x]$. Let $I$ be a proper ideal of $R[x]$. We are interested in the following two questions: When is $R[x]/I$ Frobenius? When is $R[x]/I$ Frobenius and local? We develop algorithms for answering both questions. When the nilpotency of $\text{rad}\,R$ is small, the algorithms provide explicit answers to the questions.
Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of quasi $J$-ideal which is a generalization of $J$-ideal. A proper ideal of $R$ is called a quasi $J$-ideal if its radical is a $J$-ideal. Many characterizations of quasi $J$-ideals in some special rings are obtained. We characterize rings in which every proper ideal is quasi $J$-ideal. Further, as a generalization of presimplifiable rings, we define the notion of quasi presimplifiable rings. We call a ring $R$ a quasi presimplifiable ring if whenever $a,b\in R$ and $a=ab$, then either $a$ is a nilpotent or $…
Let $\mathcal{I}(R)$ be the set of all ideals of a ring $R$, $δ$ be an expansion function of $\mathcal{I}(R)$. In this paper, the $δ$-$J$-ideal of a commutative ring is defined, that is, if $a, b\in R$ and $ab\in I\in \mathcal{I}(R)$, then $a\in J(R)$ (the Jacobson radical of $R$) or $b\in δ(I)$. Moreover, some properties of $δ$-$J$-ideals are discussed,such as localizations, homomorphic images, idealization and so on.
Quasi sdf-absorbing ideals in commutative rings
本文研究准sdf吸收理想，探讨其在局部化、满射像等构造下的稳定性，并讨论其素性条件及在整数环中的分类。
Square-difference factor absorbing ideals of a commutative ring
本文定义并研究交换环中由平方差条件刻画的 sdf-absorbing 理想。
Let $R$ be a commutative ring with identity and $S$ a multiplicatively closed subset of $R$. This paper aims to introduce the concept of $S$-$n$-ideals as a generalization of $n$-ideals. An ideal $I$ of $R$ disjoint with $S$ is called an $S$-$n$-ideal if there exists $s\in S$ such that whenever $ab\in I$ for $a,b\in R$, then $sa\in\sqrt{0}$ or $sb\in I$. The relationship among $S$% -$n$-ideals, $n$-ideals, $S$-prime and $S$-primary ideals are clarified. Several properties, characterizations and examples are presented such as investigating $S$-$n$-ideals under various contexts of constructions …
Let $R$ be a commutative ring with nonzero identity. A. Yassine et al. defined in the paper (Yassine, Nikmehr and Nikandish, 2020), the concept of $1$-absorbing prime ideals as follows: a proper ideal $I$ of $R$ is said to be a $1$-absorbing prime ideal if whenever $xyz\in I$ for some nonunit elements $x,y,z\in R$, then either $xy\in I$ or $z\in\ I$. We use the concept of $1$-absorbing prime ideals to study those commutative rings in which every proper ideal is a product of $1$-absorbing prime ideals (we call them $OAF$-rings). Any $OAF$-ring has dimension at most one and local $OAF$-domains $…
We use the concept of 2-absorbing ideal introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local TAF-domains are the atomic pseudo-valuation domains.
Freiman's theorem gives a lower bound for the cardinality of the doubling of a finite set in $\RR^n$. In this paper we give an interpretation of his theorem for monomial ideals and their fiber cones. We call a quasi-equigenerated monomial ideal a Freiman ideal, if the set of its exponent vectors achieves Freiman's lower bound for its doubling. Algebraic characterizations of Freiman ideals are given, and finite simple graphs are classified whose edge ideals or matroidal ideals of its cycle matroids are Freiman ideals.
The support of any module over a commutative ring is defined as the collection of all prime ideals of the ring at which the localization of the module is non-zero. For finitely generated modules, the support is the collection of all prime ideals containing the annihilator of the module. In this article, we raise the natural question that over which commutative rings, the support of every module is the collection of all the prime ideals of its annihilator. We completely classify such rings, and in the process it also comes out that it is enough to require that only for countably generated modul…
All rings are commutative with $1$ and $n$ is a positive integer. Let $ϕ: J(R)\to J(R)\cup{\emptyset}$ be a function where $J(R)$ denotes the set of all ideals of $R$. We say that a proper ideal $I$ of $R$ is $ϕ$-$n$-absorbing primary if whenever $a_1,a_2,...,a_{n+1}\in R$ and $a_1a_2\cdots a_{n+1}\in I\backslashϕ(I)$, either $a_1a_2\cdots a_n\in I$ or the product of $a_{n+1}$ with $(n-1)$ of $a_1,...,a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of $ϕ$-$n$-absorbing primary ideals.
Golod ideals in combinatorial commutative algebra
本文研究了标准格雷度代数的Golod性质，证明了行列式理想、二项式边理想和 permanental 理想当且仅当具有线性解时才是Golod的。接着，作者通过考虑Koszul循环的多度和Massey乘积，给出了覆盖理想定义Golod环的条件。最后，证明了无平方强Golod理想（更一般地，lcm-强Golod理想）是Golod的，而不仅仅是弱Golod。
Nil-prime ideals of a commutative ring
本文在交换环中引入幂零素理想概念以推广素理想，并研究幂零极大理想、幂零极小理想、幂零主理想等概念及其相应性质。
Maximal Ideals in Commutative Rings and the Axiom of Choice
本文给出选择公理蕴含极大理想定理的逆向推导，旨在提供自洽且易于非专家理解的证明。
Demotions of ideals in commutative rings with applications to normally torsion-freeness
本文在交换 Noether 环尤其是多项式环中引入并研究理想的降级概念，揭示其与正规无挠性的关系，并比较理想的降级与约化。
An ideal I of a commutative ring R is said to be irreducible if it cannot be written as the intersection of two larger ideals. A proper ideal I of a ring R is said to be strongly irreducible if for each ideals J, K of R, J\cap K\subseteq I implies that J\subset I or K\subset I. In this paper, we introduce the concepts of 2-irreducible and strongly 2-irreducible ideals which are generalizations of irreducible and strongly irreducible ideals, respectively. We say that a proper ideal I of a ring R is 2-irreducible if for each ideals J, K and L of R, I= J\cap K\cap L implies that either I=J\cap K …
We determine the metric dimension of the annihilating-ideal graph of a local finite commutative principal ring and a finite commutative principal ring with two maximal ideals. We also find the bounds for the metric dimension of the annihilating-ideal graph of an arbitrary finite commutative principal ring.
Some results on the ideal-based cozero-divisor graph of a commutative ring
本文研究含单位元交换环R关于理想I的余零因子图Γ''_I(R)，给出了该图的一些性质结果。
Let R be a ring and let G be a group. We prove a rather curious necessary and sufficient condition for the commutative group ring RG to be weakly nil-neat only in terms of R,G and their sections. This somewhat expands three recent results, namely those established by McGovern et al. in (J. Algebra Appl., 2015), by Danchev-McGovern in (J. Algebra, 2015) and by the present authors in (J. Taibah Univ. Sci., 2020), related to commutative nil-clean, weakly nil-clean and nil-neat group rings, respectively.
Prime ideals in infinite products of commutative rings
该研究刻画了交换环无限乘积中的素理想与极大理想，证明极大理想由特定布尔代数上的超滤子诱导，并针对两类环族完全刻画了对应乘积环的极大或全部素理想。
New Results On $S-r-$ideals in Commutative Rings
本文研究了交换环中 $S-r-$理想 的基本性质与刻画，定义了 $S-uz-$环 并给出其等价条件，同时在融合扩张、平凡扩张及多项式环中探讨了该理想的结构。
Some factorization properties of idealization in commutative rings with zero divisors
本文研究交换环中模的理想化的分解性质，证明 ACCP 的等价条件，说明 BF 性质不一定保持并给出保持条件，同时刻画了理想化环为 UFR 的情形。
In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties under homomorphic image and their transfer to various contexts of constructions such as direct product, trivial ring extension and amalgamated duplication of a ring along an ideal. Our results generate examples that enrich the current literature with new and original families of rings satisfying these properties.
It is proved that for a ring $R$ that is either an affine algebra over a field, or an equicharacteristic complete local ring, some power of the Jacobian ideal of $R$ annihilates $\mathrm{Ext}^{d+1}_{R}(-,-)$, where $d$ is the Krull dimension of $R$. Sufficient conditions are identified under which the Jacobian ideal itself annihilates these Ext-modules, and examples are provided that show that this is not always the case. A crucial new idea is to consider a derived version of the Noether different of an algebra.
In the first chapter we present new results related on monomial ideals of Borel type. Also, we introduce a new class of monomial ideals, called $\de$-fixed ideals, which generalize the class of $p$-Borel ideals and we extend several results to this new class. In the second chapter, we compute the generic initial ideal, with repect to the reverse lexicographic order, of an ideal which define a complete intersection of embedding dimension three with strong Lefschetz property and we show that it is an almost reverse lexicographic ideal. This enable us to give a proof for Moreno's conjecture in th…
A Study of S-primary Ideals in Commutative Semirings
本文研究了交换半环中S-素理想的概念、性质及其分解结构，证明了相关存在性和唯一性定理，并探讨了主理想半域中S-素理想的表现形式。
The $i$-extended ideal-based cozero-divisor graph of a commutative ring
本文研究了交换环的i-扩展理想基余零除数图，定义了图的顶点和边的条件，并探讨了其结构性质。
Countable strongly annihilated ideals in commutative rings
本文研究交换环（尤其是连续函数环）中的可数强零化理想，刻画其与实极大$z^\circ$-理想、几乎$P$-空间、Lindelöf空间的等价关系，并否定回答了Gilmer和McAdam的一个问题。
Let $R$ be a commutative ring with $ 1 \neq 0$. We recall that a proper ideal $I$ of $R$ is called a semiprimary ideal of $R$ if whenever $a,b\in R$ and $ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. We say $I$ is a {\it weakly semiprimary ideal} of $R$ if whenever $a,b\in R$ and $0 \not = ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. In this paper, we introduce a new class of ideals that is closely related to the class of (weakly) semiprimary ideals. Let $I(R)$ be the set of all ideals of $R$ and let $δ: I(R) \rightarrow I(R)$ be a function. Then $δ$ is called an expansion functio…
Let $R$ be a commutative ring and $M$ be an $R$-module, and let $I(R)^*$ be the set of all non-trivial ideals of $R$. The $M$-intersection graph of ideals of $R$, denoted by $G_M(R)$, is a graph with the vertex set $I(R)^*$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IM\cap JM\neq 0$. For every multiplication $R$-module $M$, the diameter and the girth of $G_M(R)$ are determined. Among other results, we prove that if $M$ is a faithful $R$-module and the clique number of $G_M(R)$ is finite, then $R$ is a semilocal ring. We denote the $\mathbb{Z}_n$-intersection graph of i…
A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in [M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi, Commutative Noetherian local rings whose ideals are direct sums of cyclic modules, J. Algebra 345 (2011) 257--265] the authors considered this question in the context of finite direct products of commutative Noetherian lo…
Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if for all nonunit elements $a,b,c \in R$ such that $0\neq abc \in I$, then either $ab \in I$ or $c \in I$. A number of results concerning weakly $1$-absorbing prime ideals and examples of weakly $1$-absorbing prime ideals are given. It is proved that if $I$ is a weakly $1$-absorbing prime ideal of a ring $R$ and $0 \neq I_1I_2I_3 \subseteq I$ for some ideals $I_…
This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $A$ be a commutative ring with a nonzero identity $1\neq 0$. A proper ideal $P$ of $A$ is said to be a weakly 1-absorbing prime ideal if for each nonunits $x, y, z \in A$ with $0\neq xyz \in P$, then either $xy \in P$ or $z \in P$. In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in $C(X)$, which is the ring of continuous f…
The prime ideal sum graph of a commutative unital ring $R$, denoted by $PIS(R)$, is an undirect and simple graph whose vertices are non-trivial ideals of $R$ and there exists and edge between to distinct vertices if and only if their sum is a prime ideal of $R$. In this paper, the metric dimension of $PIS(R)$ is discussed and some formulae for this parameter in $PIS(R)$ are given.
We consider the class of all commutative reduced rings for which there exists a finite subset T of A such that all projections on quotients by prime ideals of A are surjective when restricted to T. A complete structure theorem is given for this class of rings, and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals.
We establish a link between trace modules and rigidity in modules over Noetherian rings. Using the theory of trace ideals we make partial progress on a question of Dao, and on the Auslander-Reiten conjecture over Artinian Gorenstein rings.
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