On rings with divided nil ideal: a survey

arXiv (Cornell University), 2017

We introduce a new class of commutative rings with unity, namely, the Containment-Division Rings (CDR-s). We show that this notion has a very exceptional origin, since it was essentially co-discovered with the qualitative help of a computer program (i.e. The Heterogeneous Tool Set (HETS)). Besides, we show that in a Noetherian setting, the CDR-s are just another way of describing Dedekind domains. Simultaneously, we see that for CDR-s, the Noetherian condition can be replaced by a weaker Divisor Chain Condition.

Lecture notes in pure and applied mathematics, 2005

A commutative ring R is said to be a φ-ring if its nilradical N il(R) is both prime and divided, the latter meaning N il(R) is comparable with each principal ideal of R. Special types include φ-Noetherian (also known as nonnil-Noetherian), φ-Mori, φ-chained and φ-Prüfer. A ring R is φ-Noetherian if N il(R) is a divided prime and each ideal that properly contains N il(R) is finitely generated. If R is a φ-Noetherian ring and X 1 , X 2 ,. .. , X n are indeterminates, then an ideal I of R[X 1 , X 2 ,. .. , X n ] which contains a nonnil element of R is finitely generated. Also, for a ring R where N il(R) is a nonzero prime ideal with N il(R) 2 = (0), there is a ring A whose nilradical N il(A)

Communications in Algebra, 2001

Journal of Mathematics of Kyoto University

2004

Throughout, R will be a commutative ring with identity with total quotient ring T (R), group of units U (R), set of zero-divisors Z(R), and Jacobson radical J(R). For a; b 2 R, we de…ne three associate relations:

Houston journal of mathematics

The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Prüfer domains and Bezout domains. Let H = {R | R is a commutative ring with 1 = 0 and N il(R) is a divided prime ideal of R}. Let R ∈ H, T (R) be the total quotient ring of R, and set φ : T (R) −→ R N il(R) such that φ(a/b) = a/b for every a ∈ R and b ∈ R \ Z(R). Then φ is a ring homomorphism from T (R) into R N il(R) , and φ restricted to R is also a ring homomorphism from R into R N il(R) given by φ(x) = x/1 for every x ∈ R. A nonnil ideal I of R is said to be φ-invertible if φ(I) is an invertible ideal of φ(R). If every finitely generated nonnil ideal of R is φ-invertible, then we say that R is a φ-Prüfer ring. Also, we say that R is a φ-Bezout ring if φ(I) is a principal ideal of φ(R) for every finitely generated nonnil ideal I of R. We show that the theories of φ-Prüfer and φ-Bezout rings resemble that of Prüfer and Bezout domains.

2011

Let D be an integral domain with quotient field K. The b-operation that associates to each nonzero D-submodule E of K, E^b := {EV | V valuation overring of D}, is a semistar operation that plays an important role in many questions of ring theory (e.g., if I is a nonzero ideal in D, I^b coincides with its integral closure). In a first part of the paper, we study the integral domains that are b-Noetherian (i.e., such that, for each nonzero ideal I of D, I^b = J^b for some a finitely generated ideal J of D). For instance, we prove that a b-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily b-Noetherian. We also characterize several distinguished classes of b-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type _a canonically associated to a given semistar operation (for instance, the b-operation is th...

Open Mathematics, 2023

In this article, we are interested in uniformly pr-ideals with order ≤2 (which we call r 2-ideals) introduced by Rabia Üregen in [On uniformly pr-ideals in commutative rings, Turkish J. Math. 43 (2019), no. 4, 18781886]. Several characterizations and properties of these ideals are given. Moreover, the comparison between the (nonzero) r 2-ideals and certain classes of classical ideals gives rise to characterizations of certain rings based only on the properties of the ideals consisting only of zero-divisors. Namely, among other things, we compare the class of (nonzero) r 2-ideals with the class of (minimal) prime ideals, the class of minimal prime ideals and their squares, and the class of primary ideals. The study of r 2-ideal in polynomial rings allows us to give a new characterization of the rings satisfying the famous A-property.

Bulletin of the Australian Mathematical Society, 1981

Bulletin of the Australian Mathematical Society, 1980