WPZI RINGS AND STRONG REGULARITY

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Junchao Wei
Keyword(s):  
Regular Ring ◽  
Strong Regularity ◽  
Strongly Regular

Abstract In this paper, we study the strong regularity of left SF rings and obtain the following results: Let R be a left SF ring. If R satisfies one of the following conditions, then R is a strongly regular ring: 1) R is a left WPZI ring; 2) R is a right WPZI ring; 3) R is a right weakly semicommutative ring; 4) R is a semicommutative ring; 5) R is a reversible ring.

On a Sheaf of Division Rings*

1975 ◽  
Vol 17 (5) ◽  
pp. 727-731
Author(s):  
George Szeto
Keyword(s):  
Principal Ideal ◽  
Regular Ring ◽  
Central Idempotent ◽  
Division Rings ◽  
Maximal Ideals ◽  
Strongly Regular ◽  
Th 1

R. Arens and I. Kaplansky ([1]) call a ring A biregular if every two sided principal ideal of A is generated by a central idempotent and a ring A strongly regular if for any a in A there exists b in A such that a=a2b. In ([1], Sections 2 and 3), a lot of interesting properties of a biregular ring and a strongly regular ring are given. Some more properties can also be found in [3], [5], [8], [9] and [13]. For example, J. Dauns and K. Hofmann ([3]) show that a biregular ring A is isomorphic with the global sections of the sheaf of simple rings A/K where K are maximal ideals of A. The converse is also proved by R. Pierce ([9], Th. 11–1). Moreover, J. Lambek ([5], Th. 1) extends the above representation of a biregular ring to a symmetric module.

scholarly journals Strong Regularity in Arbitrary Rings

1952 ◽  
Vol 4 ◽  
pp. 51-53 ◽  
Author(s):  
Tetsuo Kandô
Keyword(s):  
Matrix Ring ◽  
Strong Regularity ◽  
Full Matrix ◽  
Regular Ideal ◽  
Strongly Regular ◽  
Full Matrix Ring

An element a of a ring R is called regular, if there exists an element x of R such that a×a = a, and a two-sided ideal a in R is said to be regular if each of its elements is regular B. Brown and N. H. McCoy [1] has recently proved that every ring R has a unique maximal regular two-sided ideal M(R), and that M(R) has the following radical-like property: (i) M(R/M(R)) = 0; (ii) if a is a two-sided ideal of R, then M(a) = a ∩ M(R); (iii) M(Rn) = (M(R))n, where Rn denotes a full matrix ring of order n over R. Arens and Kaplansky [2] has defined an element a of R to be strongly regular when there exists an element x of R such that a2x = a. We shall prove in this note that replacing “regularity” by “strong regularity,” we have also a unique maximal strongly regular ideal N(R), and shall investigate some of its properties.

scholarly journals Strongly regular points of mappings

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Malek Abbasi ◽  
Michel Théra
Keyword(s):  
Error Bound ◽  
Directional Derivative ◽  
Sufficient Conditions ◽  
Strong Regularity ◽  
Strongly Regular

AbstractIn this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound.

scholarly journals Positive derivations on partially ordered strongly regular rings

1984 ◽  
Vol 37 (2) ◽  
pp. 178-180 ◽  
Author(s):  
D. J. Hansen
Keyword(s):  
Regular Ring ◽  
Additional Property ◽  
Partially Ordered ◽  
Strongly Regular ◽  
Regular Rings

AbstractThe author presents a proof that a partially ordered strongly regular ring S which has the additional property that the square of each member of S is greater than or equal to zero cannot have nontrivial positive derivations.

A note on modules over regular rings

1971 ◽  
Vol 4 (1) ◽  
pp. 57-62 ◽  
Author(s):  
K. M. Rangaswamy ◽  
N. Vanaja
Keyword(s):  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Free Left ◽  
Von Neumann Regular ◽  
Strongly Regular ◽  
Torsion Free ◽  
Regular Rings

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.

scholarly journals On strongly regular near-rings

1984 ◽  
Vol 27 (1) ◽  
pp. 61-64 ◽  
Author(s):  
Y. V. Reddy ◽  
C. V. L. N. Murty
Keyword(s):  
Strong Regularity ◽  
Strongly Regular

According to Mason [1] a right near-ring N is called (i) left (right) strongly regular if for every a there is an x in N such that a = xa2 (a = a2x) and (ii) left (right) regular if for every a there is an x in N such that a = xa2 (a = a2x) and a = axa. He proved that for a zerosymmetric near-ring with identity, the notions of left regularity, right regularity and left strong regularity are equivalent. The aim of this note is to prove that these three notions are equivalent for arbitrary near-rings. We also show that if N satisfies dec on iV-subgroups, then all the above four notions are equivalent.

Rings whose cyclics are C3-modules

2016 ◽  
Vol 15 (08) ◽  
pp. 1650152 ◽  
Author(s):  
Yasser Ibrahim ◽  
Xuan Hau Nguyen ◽  
Mohamed F. Yousif ◽  
Yiqiang Zhou
Keyword(s):  
Direct Product ◽  
Regular Ring ◽  
Matrix Ring ◽  
Artinian Ring ◽  
Local Rings ◽  
Classical Theorem ◽  
Basic Properties ◽  
Semiperfect Ring ◽  
Injectivity Condition ◽  
Strongly Regular

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or [Formula: see text]. Here we carry out a study of the rings whose cyclic modules are [Formula: see text]-modules. The motivation is the observation that a ring [Formula: see text] is semisimple artinian if and only if every [Formula: see text] -generated right [Formula: see text]-module is a [Formula: see text]-module. Many basic properties are obtained for the rings whose cyclics are [Formula: see text]-modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a [Formula: see text] matrix ring over a strongly regular ring. Applications to the rings whose [Formula: see text]-generated modules are [Formula: see text] -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed.

Morphic Properties of Extensions of Rings

2010 ◽  
Vol 17 (02) ◽  
pp. 337-344
Author(s):  
Qinghe Huang ◽  
Jianlong Chen
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Group Ring ◽  
Regular Ring ◽  
Strongly Regular ◽  
Left Annihilator

An element a in a ring R is called left morphic if R/Ra ≅ ℓR(a), where ℓR(a) denotes the left annihilator of a in R. A ring R is said to be left morphic if every element is left morphic. In this paper, it is shown that if I is an ideal of a unit regular ring R, then for each positive integer n, [Formula: see text] is a left morphic ring. This extends two recent results of Lee and Zhou. It is also proved that if R is a strongly regular ring and Cn= 〈g〉 is a cyclic group of order n ≥ 2, then for any r ∈ R, 1 + rg is morphic in the group ring RCn.

scholarly journals Some characterizations of π-regular rings with no infinite trivial subring

1991 ◽  
Vol 34 (1) ◽  
pp. 1-5
Author(s):  
Yasuyuki Hirano
Keyword(s):  
Commutative Ring ◽  
Regular Ring ◽  
Finite Ring ◽  
Periodic Ring ◽  
Strongly Regular ◽  
Regular Rings

It is shown that a ringRis a π-regular ring with no infinite trivial subring if and only ifRis a subdirect sum of a strongly regular ring and a finite ring. Some other characterizations of such a ring are given. Similar result is proved for a periodic ring. As a corollary, it is shown that every δ-ring is a subdirect sum of a Unite ring and a commutative ring. This was conjectured by Putcha and Yaqub.

scholarly journals Slices, RNP, strong regularity, and martingales

1990 ◽  
Vol 41 (3) ◽  
pp. 411-415
Author(s):  
Maria Girardi ◽  
J.J. Uhl
Keyword(s):  
Simple Proof ◽  
Strong Regularity ◽  
Nikodym Property ◽  
Strongly Regular ◽  
Regular Operators

The usual proof that dentability implies the Radon-Nikodým property involves a clever but rather baroque exhaustion argument. This note presents a very short and simple proof of this implication. The techniques in this new proof are then generalised to derive some direct proofs of recent results concerning strongly regular operators on L1.

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