On Rings whose Maximal Essential Ideals are Pure (original) (raw)

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 some characterizations of regular and potent rings relative to right ideals

Novi Sad Journal of Mathematics, 2018

In this paper we study the notion of regular rings relative to right ideals, and we give another characterization of these rings. Also, we introduce the concept of an annihilator relative to a right ideal. Basic properties of this concept are proved. New results obtained include necessary and sufficient conditions for a ring to be regular (potent) relative to right ideal.

Weakly Regular Rings

Canadian Mathematical Bulletin, 1973

This paper attempts to generalize a property of regular rings, namely,I2=I for every right (left) ideal. Rings with this property are called right (left) weakly regular. A ring which is both left and right weakly regular is called weakly regular. Kovacs in [6] proved that, for commutative rings, weak regularity and regularity are equivalent conditions and left open the question whether for arbitrary rings the two conditions are equivalent. We show in §1 that, in general weak regularity does not imply regularity. In fact, the class of weakly regular rings strictly contains the class of regular rings as well as the class of biregular rings.

On Strong (A)-Rings

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).

On adequate rings

Journal of Taibah University for Science, 2015

In this paper, we investigate the transfer of notion of adequate rings to trivial ring extensions and pullbacks. Our aim is to give new classes of commutative rings satisfying this property.

On Continuous Rings

Journal of Algebra, 1997

We show that if R is a semiperfect ring with essential left socle and rl K s K for every small right ideal K of R, then R is right continuous. Accordingly some well-known classes of rings, such as dual rings and rings all of whose cyclic right R-modules are essentially embedded in projectives, are shown to be continuous. We also prove that a ring R has a perfect duality if and only if the dual of every simple right R-module is simple and R [ R is a left and right CS-module. In Sect. 2 of the paper we provide a characterization for semiperfect right self-injective rings in terms of the CS-condition.

Some results on N-pure ideals

Cornell University - arXiv, 2022

In this paper, we consider the N-pure notion. An ideal I of a ring R is said to be N-pure, if for every a ∈ I there exists b ∈ I such that a(1 − b) ∈ N (R), where N(R) is nil radical of R. We provide new characterizations for N-pure ideals. In addition, N-pure ideals of an arbitrary ring are identified. Also, some other properties of N-pure ideals are studied. finally, we prove some results about the endomorphism ring of pure and N-pure ideals.

A note on modules over regular rings

Bulletin of the Australian Mathematical Society

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

On strongly prime rings and ideals

Communications in Algebra, 2000

Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.