Some results on N-pure ideals (original) (raw)
arXiv: Commutative Algebra, 2020
In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it mid ring. Also, we provide new characterizations for von Neumann regular and zero-dimensional rings. Moreover, some results about mp-ring are given. Finally, a characterization for mid rings is provided. Then it is shown that the class of mid rings is strictly between the class of reduced mp-rings (p.f. rings) and the class of mp-rings.
2012
In this paper, we introduced the notion of N-ideals on rings and investigated some related properties
On Rings whose Maximal Essential Ideals are Pure
AL-Rafidain Journal of Computer Sciences and Mathematics, 2007
This paper introduces the notion of a right MEP-ring (a ring in which every maximal essential right ideal is left pure) with some of their basic properties; we also give necessary and sufficient conditions for MEPrings to be strongly regular rings and weakly regular rings.
Maximal Generalization of Pure Ideals
2008
The purpose of this paper is to study the class of the rings for which every maximal right ideal is left GP-ideal. Such rings are called MGP-rings and give some of their basic properties as well as the relation between MGP-rings, strongly regular ring, weakly regular ring and kasch ring.
On n-absorbing ideals and (m,n)-closed ideals in trivial ring extensions of commutative rings
Journal of Algebra and Its Applications
Let [Formula: see text] be a commutative ring with [Formula: see text]. Recall that a proper ideal [Formula: see text] of [Formula: see text] is called a 2-absorbing ideal of [Formula: see text] if [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text]. A more general concept than 2-absorbing ideals is the concept of [Formula: see text]-absorbing ideals. Let [Formula: see text] be a positive integer. A proper ideal [Formula: see text] of [Formula: see text] is called an n-absorbing ideal of [Formula: see text] if [Formula: see text] and [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text]. The concept of [Formula: see text]-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of [Formula: see text] is a 1-absorbing ideal of [Formula: see text]). Let [Formula: see text] and [Formula: see text] be integers with [Fo...
Mediterranean Journal of Mathematics, 2012
In this paper, we introduce a strong property (A) as follows: A ring R is called satisfying strong property (A) if every finitely generated ideal of R which is generated by a finite number of zero-divisors elements of R, has a non zero annihilator. We study the transfer of property (A) and strong property (A) in trivial ring extensions and amalgamated duplication of a ring along an ideal. We also exhibit a class of rings which satisfy property (A) and do not satisfy strong property (A).
n-absorbing I-primary ideals in commutative rings
arXiv (Cornell University), 2022
We define a new generalization of −absorbing ideals in commutative rings called −absorbing −primary ideals. We investigate some characterizations and properties of such new generalization. If is an −absorbing −primary ideal of and √ = √ , then √ is a −absorbing −primary ideal of. And if √ is an (− 1) −absorbing ideal of such that � √ ⊆ , then is an −absorbing −primary ideal of .
Formalized Mathematics, 2021
Summary. We formalize in the Mizar System [3], [4], definitions and basic propositions about primary ideals of a commutative ring along with Chapter 4 of [1] and Chapter III of [8]. Additionally other necessary basic ideal operations such as compatibilities taking radical and intersection of finite number of ideals are formalized as well in order to prove theorems relating primary ideals. These basic operations are mainly quoted from Chapter 1 of [1] and compiled as preliminaries in the first half of the article.
Some Results on Normal Homogeneous Ideals
Communications in Algebra, 2003
In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an Ngraded ring A of the form A ≥m := ℓ≥m A ℓ and monomial ideals in a polynomial ring over a field. For ideals of the form A ≥m we generalize a recent result of Faridi. We prove that a monomial ideal in a polynomial ring in n indeterminates over a field is normal if and only if the first n− 1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals of the form I(λ) := J(λ), where J(λ) = (x λ1 1 ,. .. , x λn n) ⊆ K[x 1 ,. .. , x n ]. To state our main result in this setting, we let ℓ = lcm(λ 1 ,. .. , λ i ,. .. λ n), for 1 ≤ i ≤ n, and set λ ′ = (λ 1 ,. .. , λ i−1 , λ i + ℓ, λ i+1 ,. .. , λ n). We prove that if I(λ ′) is normal then I(λ) is normal and that the converse holds with a small additional assumption.