Related papers: Absorbing ideals: a survey on $\omega$-stable grou…
We show that any $n$-absorbing ideal must be strongly $n$-absorbing, which is the first of Anderson and Badawi's three interconnected conjectures on absorbing ideals. We prove this by introducing and studying objects called maximal and…
All rings are commutative with $1$ and $n$ is a positive integer. Let $\phi: J(R)\to J(R)\cup{\emptyset}$ be a function where $J(R)$ denotes the set of all ideals of $R$. We say that a proper ideal $I$ of $R$ is $\phi$-$n$-absorbing primary…
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…
Motivated by situations in which the removal of a zero (a.k.a., an absorbing element) from a semigroup yields a subsemigroup with another zero, sets of quasi-zeros (a.k.a., quasi-absorbing elements) are introduced as well as primitive…
A commutative ring $R$ is stable if every non-zero ideal $I$ of $R$ is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much…
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.…
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…
In this article, we introduce and study the concept of $\phi$-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…
Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these…
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…
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…
The present paper addresses the notion of $(u,v)$-absorbing primary hyperideals in commutative multiplicative hyperrings.
We introduce and prove the consistency of a new set theoretic axiom we call the \emph{Invariant Ideal Axiom}. The axiom enables us to provide (consistently) a full topological classification of countable sequential groups, as well as fully…
We introduce and study varions notions of completeness of translation-invariant ideals in groups.
We improve homological stability ranges for the orthogonal group, special orthogonal group, elementary orthogonal group and the spin group over a commutative local ring $R$ with infinite residue field such that $2 \in R^{*}$.
In this paper, we introduce and study two new classes of commutative rings, namely semi transitional rings and transitional rings, which extend several classical ideas arising from rings of continuous functions and their variants. A general…
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…
We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these…
We introduce the notion of an R-group of which the clas- sical groups R, Z and R_+ are typical examples, and we study flows (X;H), where X is a locally compact space and H is a continuous R- group action on X with the further property that…
Irreducible decompositions of monomial ideals in polynomial rings over a field are well-understood. In this paper, we investigate decompositions in the set of monomial ideals in the semigroup ring A[\mathbb{R}_{\geq 0}^d] where A is an…