On 1-absorbing primary ideal of a commutative ring (Correction to Theorem 17 is added)

On Graded classical 2-absorbing second submodules of graded modules over graded commutative rings

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.

交换乘法超环中的(u,v)-吸收（素）超理想

(u,v)-absorbing (prime) hyperideals in commutative multiplicative hyperrings

该研究在交换乘法超环框架下提出(u,v)-吸收超理想概念并探究其性质，进一步拓展出(u,v)-吸收素超理想的概念并推导相关结论，丰富了超环理想理论。

交换环上模的广义平方差因子吸收子模

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。

交换环的平方差因子吸收理想

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。

准sdf吸收理想在交换环中的研究

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。

Classical 2-absorbing submodules of modules over commutative rings

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 $bcm\in N$.

On ϕ-n-absorbing primary ideals of commutative rings

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.

具有平方差因子吸收分解的交换环

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。

On 2-absorbing primary submodules of modules over commutative rings

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…

On graded $A$-2-absorbing submodules of graded modules over graded commutative rings

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.

On Weakly 1-Absorbing Prime Ideals

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…

Steps in Anderson-Badawis Conjecture on n-Absorbing and Strongly n-Absorbing Ideals

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

乘法超环中的$(u,v)$-吸收准素超理想

$(u,v)$-absorbing primary hyperideals in multiplicative hyperrings

本文在交换乘法超环中引入并研究了$(u,v)$-吸收准素超理想的概念，拓展了超环理想理论。

On weakly $1$-absorbing prime ideals of commutative rings

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_…

On $ϕ$-1-Absorbing Prime Ideals

In this paper, we introduce $ϕ$-1-absorbing prime ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity $1\neq0$ and $ϕ:\mathcal{I}(R)\rightarrow\mathcal{I}(R)\cup\{\emptyset\}$ be a function where $\mathcal{I}(R)$ is the set of all ideals of $R$. A proper ideal $I$ of $R$ is called a $ϕ$-1-absorbing prime ideal if for each nonunits $x,y,z\in R$ with $xyz\in I-ϕ(I)$, then either $xy\in I$ or $z\in I$. In addition to give many properties and characterizations of $ϕ$-1-absorbing prime ideals, we also determine rings in which every proper ideal is $ϕ$-1-absorbing prim…

交换环中的n-吸收理想分解

n-absorbing ideal factorization in commutative rings

本文揭示了交换环理论中Mori domains、pseudo-valuation domains与$n$-absorbing ideals的内在联系，证明了整环的真理想可分解为有限个$n$-吸收理想乘积的充要条件，并给出了具体环例与二次数域的刻画。

The radical of an n-absorbing ideal

In this note we show that in a commutative ring $R$ with unity, for any $n > 0$, if $I$ is an $n$-absorbing ideal of $R$, then $(\sqrt{I})^{n} \subseteq I$.

Generalization of 2-absorbing quasi primary ideals

In this article, we introduce and study the concept of $ϕ$-2-absorbing quasi primary ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity 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 $ϕ$-2-absorbing quasiprimary ideal of $R$ if $a,b,c\in R$ and whenever $abc\in I-ϕ(I),$ then either $ab\in\sqrt{I}$ or $ac\in\sqrt{I}$ or $bc\in\sqrt{I}$. In addition to giving many properties of $ϕ$-2-absorbing quasi primary ideals, we also use them to character…

2-absorbing and strongly 2-absorbing secondary submodules of modules

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.

Strongly $ψ$-$2$-absorbing second submodules

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.

$S$-2-absorbing submodules and $S$-2-absorbing second submodules

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.

On 2-Absorbing Primary Hyperideals Of Multiplicative Hyperrings

Primary hyperideals have been introduced and studied in multiplicative hyperrings. In this paper, we intend to study extensively primary hyperideals of multiplicative hyperrings with absorbing zero and prove some results regarding them. Also, we describe Cu- ideals of multiplicative hyperrings which are particular classes of hyperideals. In the last section, we introduce 2-absorbing primary hyperideals and investigate the properties of this notion in commutative multiplicative hyperrings.

关于1-吸收素子半模与弱1-吸收素子半模

On 1-absorbing prime and weakly 1-absorbing prime subsemimodules

本文在交换半环上引入1-吸收素与弱1-吸收素子半模概念，并给出其理想与子半模乘积形式的等价刻画。

Some properties of graded generalized 2-absorbing submodules

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.

吸收理想：关于 $\omega$-稳定群的综述

Absorbing ideals: a survey on $ω$-stable groups

本文在交换环吸收理想理论中引入 $\omega$-稳定群，通过正反例刻画其性质，建立其与 H-同余的关系，并研究传递性条件下的群作用。

1-absorbing primary submodules

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.

$n$-absorbing monomial ideals in polynomial rings

In a commutative ring $R$ with unity, given an ideal $I$ of $R$, Anderson and Badawi in 2011 introduced the invariant $ω(I)$, which is the minimal integer $n$ for which $I$ is an $n$-absorbing ideal of $R$. In the specific case that $R = k[x_{1}, \ldots, x_{n}]$ is a polynomial ring over a field $k$ in $n$ variables $x_{1},\ldots, x_{n}$, we calculate $ω(I)$ for certain monomial ideals $I$ of $R$.

理想的集合：任意$n$-吸收理想都是强$n$-吸收的

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。

具有 $n$-$1$-吸收素素因子分解的交换环

Commutative rings with $n$-$1$-absorbing prime factorization

本文研究了具有 $n$-$1$-吸收素素因子分解的交换环的性质，并探讨了其在多项式环、幂级数环等扩展中的应用。

On 1-absorbing prime submodules

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…

关于 $S$-$n$-吸收理想

On $S$-$n$-absorbing ideals

本文在交换环中引入了 $S$-$n$-吸收理想的概念，作为 $S$-素理想和 $n$-吸收理想的推广，并研究了其基本性质、商扩张、存在性与合并。

On (k,n)-closed 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.

自由交换幺半群的理想扩张

Ideal extensions of free commutative monoids

本文引入间隙吸收幺半群，证明其链次度至多四且长度集为区间，并给出理想扩张的ω-素性界及有限间隙的等价条件。

On Graded Strongly $1$-Absorbing Primary Ideals

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…

On Graded $ϕ$-$1$-absorbing prime ideals

Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $ϕ:GI(R)\rightarrow GI(R)\cup\{\emptyset\}$ is a function. In this article, we introduce and study the concept of graded $ϕ$-$1$-absorbing prime ideals. A proper graded ideal $I$ of $R$ is called a graded $ϕ$% -$1$-absorbing prime ideal of $R$ if whenever $a,b,c$ are homogeneous nonunit elements of $R$ such that $abc\in I-ϕ(I)$, then $ab\in I$ or $c\in I$. Several properties of graded $ϕ$-$1$-absorbing prime ideals have been examined.

On Graded $1$-Absorbing Prime Submodules

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…

关于近似理想：描述性关系子空间中的准素理想与1-吸收理想

On approximately ideals: primary and 1-absorbing ideals in descriptive relator spaces

本文在描述性关系子空间中研究近似交换环的近似理想，提出了近似准素、半准素及1-吸收准素理想的概念与定理，并揭示了这些理想类之间的包含与转化关系。

On $1$-absorbing $δ$-primary ideals

Let $R$ be a commutative ring with nonzero identity. Let $\mathcal{I}(R)$ be the set of all ideals of $R$ and let $δ: \mathcal{I}(R)\longrightarrow \mathcal{I}(R)$ be a function. Then $δ$ is called an expansion function of ideals of $R$ if whenever $L, I, J$ are ideals of R with $J \subseteq I$, we have $L \subseteq δ( L)$ and $δ(J)\subseteq δ(I)$. Let $δ$ be an expansion function of ideals of $R$. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of $δ$-primary ideals. A proper ideal $I$ of $R$ is said to be a $1$-absorbing $δ$-primary idea…

借助若干类型模刻画弱伪准素2-吸收子模

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。

关于2-吸收子模及其融合

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。

On Generalizations of Graded $2$-absorbing and Graded $2$-absorbing primary submodules

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 …

On the Anderson-Badawi $ω_{R[X]}(I[X])=ω_R(I)$ conjecture

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1 \cdots x_{n+1} \in I$ for $x_1, \ldots, x_{n+1} \in R$, then there are $n$ of the $x_i$'s whose product is in $I$ and conjecture that $ω_{R[X]}(I[X])=ω_R(I)$ for any ideal $I$ of an arbitrary ring $R$, where $ω_R(I)= \min \{n\colon\text{$I$ is an $n$-absorbing ideal of $R$}\}$. In the present paper, we use content formula techniques …

Hilbert Coefficients of Parameter Ideals

We consider the non-positivity of the Hilbert coefficients for a parameter ideal of a commutative Noetherian local ring. In particular, we show that the second Hilbert coefficient of a parameter ideal of depth at least d-1 is always non-positive and give a condition for the coefficient to be zero. With the added condition that the depth of the associated graded ring is also at least d- we show $e_{i}(q) \leq 0$ for $i=1, ..., d$.

Hilbertian Frobenius algebras

Commutative Hilbertian Frobenius algebras are those commutative semi-group objects in the monoidal category of Hilbert spaces, for which the Hilbert adjoint of the multiplication satisfies the Frobenius compatibility relation, that is, this adjoint is a bimodule map. In this note we prove that they split as an orthogonal direct sum of two closed ideals, their Jacobson radical which in fact is nothing but their annihilator, and the closure of the linear span of their group-like elements. As a consequence such an algebra is semisimple if, and only if, its multiplication has a dense range. In par…