scholarly journals On weakly clean and weakly exchange rings having the strong property

2017 ◽  
Vol 101 (115) ◽  
pp. 135-142
Author(s):  
Peter Danchev
Keyword(s):  
Acta Math ◽  
Clean Rings ◽  
Exchange Rings ◽  
Strong Property ◽  
Regular Rings

We define two classes of rings calling them weakly clean rings and weakly exchange rings both equipped with the strong property. Although the classes of weakly clean rings and weakly exchange rings are different, their two proper subclasses above do coincide. This extends results due to W. Chen (Commun. Algebra, 2006) and Chin-Qua (Acta Math. Hungar., 2011). We also completely characterize strongly invo-regular rings, thus somewhat extending results due to Danchev-McGovern (J. Algebra, 2015). Some other principal results concerning weakly clean and weakly exchange rings are discussed as well.

Exchange rings, exchange equations, and lifting properties

2016 ◽  
Vol 26 (06) ◽  
pp. 1177-1198 ◽  
Author(s):  
Dinesh Khurana ◽  
T. Y. Lam ◽  
Pace P. Nielsen
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Square Roots ◽  
New Class ◽  
Clean Rings ◽  
Von Neumann Regular Rings ◽  
Exchange Rings ◽  
Von Neumann Regular ◽  
Regular Rings

In this paper, we study exchange rings and clean rings [Formula: see text] with [Formula: see text] (or otherwise). Analogues of a theorem of Camillo and Yu characterizing clean and strongly clean rings with [Formula: see text] are obtained for such rings (as well as for exchange rings) using the viewpoint of exchange equations introduced in a recent paper of the authors. We also study a new class of rings including von Neumann regular rings in which square roots of one (instead of idempotents) can be lifted modulo left ideals, and conjecture that such rings are exchange rings. This conjecture holds for commutative rings, and would hold for all rings if it holds for semiprimitive rings of characteristic [Formula: see text].

scholarly journals TWO QUESTIONS OF L. VAŠ ON -CLEAN RINGS

2013 ◽  
Vol 88 (3) ◽  
pp. 499-505 ◽  
Author(s):  
JIANLONG CHEN ◽  
JIAN CUI
Keyword(s):  
Regular Ring ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Clean Rings ◽  
Regular Rings

AbstractA $\ast $-ring $R$ is called (strongly) $\ast $-clean if every element of $R$ is the sum of a unit and a projection (that commute). Vaš [‘$\ast $-Clean rings; some clean and almost clean Baer $\ast $-rings and von Neumann algebras’, J. Algebra 324(12) (2010), 3388–3400] asked whether there exists a $\ast $-ring that is clean but not $\ast $-clean and whether a unit regular and $\ast $-regular ring is strongly $\ast $-clean. In this paper, we answer these two questions. We also give some characterisations related to $\ast $-regular rings.

scholarly journals A CLASS OF EXCHANGE RINGS

2008 ◽  
Vol 50 (3) ◽  
pp. 509-522 ◽  
Author(s):  
TSIU-KWEN LEE ◽  
YIQIANG ZHOU
Keyword(s):  
Exchange Ring ◽  
Glasgow Math ◽  
Basic Properties ◽  
Boolean Rings ◽  
Clean Rings ◽  
Exchange Rings

AbstractIt is well known that a ring R is an exchange ring iff, for any a ∈ R, a−e ∈ (a2−a)R for some e2 = e ∈ R iff, for any a ∈ R, a−e ∈ R(a2−a) for some e2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a−e ∈ (a2−a)R for a unique e2 = e ∈ R. This condition is not left–right symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227–236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.

On Quasipolar Rings

2012 ◽  
Vol 19 (04) ◽  
pp. 683-692 ◽  
Author(s):  
Zhiling Ying ◽  
Jianlong Chen
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Commutative Rings ◽  
Ring Theory ◽  
Clean Rings ◽  
Commutative Local Ring ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring ◽  
Regular Rings

The notion of quasipolar elements of rings was introduced by Koliha and Patricio in 2002. In this paper, we introduce the notion of quasipolar rings and relate it to other familiar notions in ring theory. It is proved that both strongly π-regular rings and uniquely clean rings are quasipolar, and quasipolar rings are strongly clean, but no two of these classes of rings are equivalent. For commutative rings, quasipolar rings coincide with semiregular rings. It is also proved that every n × n upper triangular matrix ring over any commutative uniquely clean ring or commutative local ring is quasipolar.

PSEUDOPOLAR MATRIX RINGS OVER LOCAL RINGS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350109 ◽  
Author(s):  
JIAN CUI ◽  
JIANLONG CHEN
Keyword(s):  
Positive Integer ◽  
Local Rings ◽  
Matrix Rings ◽  
Clean Rings ◽  
Regular Rings

A ring R is called pseudopolar if for every a ∈ R there exists p2 = p ∈ R such that p ∈ comm 2(a), a + p ∈ U(R) and akp ∈ J(R) for some positive integer k. Pseudopolar rings are closely related to strongly π-regular rings, uniquely strongly clean rings, semiregular rings and strongly π-rad clean rings. In this paper, we completely characterize the local rings R for which M2(R) is pseudopolar.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

On (Strongly) Gorenstein Von Neumann Regular Rings

2011 ◽  
Vol 39 (9) ◽  
pp. 3242-3252 ◽  
Author(s):  
Najib Mahdou ◽  
Mohammed Tamekkante ◽  
Siamak Yassemi
Keyword(s):  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

Coincidence and fixed points for hybrid tangential maps

2010 ◽  
Vol 17 (2) ◽  
pp. 273-285
Author(s):  
Tayyab Kamran ◽  
Quanita Kiran
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Property ◽  
Fixed Point Theorems ◽  
Acta Math ◽  
Contractive Condition ◽  
The Common

Abstract In [Int. J. Math. Math. Sci. 2005: 3045–3055] by Liu et al. the common property (E.A) for two pairs of hybrid maps is defined. Recently, O'Regan and Shahzad [Acta Math. Sin. (Engl. Ser.) 23: 1601–1610, 2007] have introduced a very general contractive condition and obtained some fixed point results for hybrid maps. We introduce a new property for pairs of hybrid maps that contains the property (E.A) and obtain some coincidence and fixed point theorems that extend/generalize some results from the above-mentioned papers.

scholarly journals The Calderón–Zygmund Theorem with an $$L^1$$ Mean Hörmander Condition

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Soichiro Suzuki
Keyword(s):  
Singular Integral ◽  
Integral Operators ◽  
Acta Math ◽  
Limited Range ◽  
Singular Integral Operators ◽  
Convolution Type ◽  
Worst Case ◽  
Hörmander Condition

AbstractIn 2019, Grafakos and Stockdale introduced an $$L^q$$ L q mean Hörmander condition and proved a “limited-range” Calderón–Zygmund theorem. Comparing their theorem with the classical one, it requires weaker assumptions and implies the $$L^p$$ L p boundedness for the “limited-range” instead of $$1< p < \infty $$ 1 < p < ∞ . However, in this paper, we show that the $$L^q$$ L q mean Hörmander condition is actually enough to obtain the $$L^p$$ L p boundedness for all $$1< p < \infty $$ 1 < p < ∞ even in the worst case $$q=1$$ q = 1 . We use a similar method to that used by Fefferman (Acta Math 124:9–36, 1970): form the Calderón–Zygmund decomposition with the bounded overlap property and approximate the bad part. Also we give a criterion of the $$L^2$$ L 2 boundedness for convolution type singular integral operators under the $$L^1$$ L 1 mean Hörmander condition.

scholarly journals Certain fundamental properties of generalized natural transform in generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Khalaf Al-Omari ◽  
Serkan Araci
Keyword(s):  
Inversion Formula ◽  
Acta Math ◽  
Generalized Functions ◽  
General Nature ◽  
Qualitative Behavior ◽  
Fundamental Properties ◽  
Integrable Functions ◽  
Extended Operator ◽  
Definition Of ◽  
Space Of Integrable Functions

AbstractThis paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

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