Von Neumann regular graphs associated with rings

2018 ◽  
Vol 10 (03) ◽  
pp. 1850029
Author(s):  
Ali Jafari Taloukolaei ◽  
Shervin Sahebi
Keyword(s):  
Regular Graph ◽  
Regular Element ◽  
Regular Graphs ◽  
Von Neumann ◽  
Basic Properties ◽  
Von Neumann Regular ◽  
Nonzero Identity

Let [Formula: see text] be a ring with nonzero identity. By the Von Neumann regular graph of [Formula: see text], we mean the graph that its vertices are all elements of [Formula: see text] such that there is an edge between vertices [Formula: see text] if and only if [Formula: see text] is a Von Neumann regular element of [Formula: see text], denoted by [Formula: see text]. In this paper, the basic properties of [Formula: see text] are investigated and some characterization results regarding connectedness, diameter, girth and planarity of [Formula: see text] are given.

A note on r-precious ring

2020 ◽  
pp. 2150231
Author(s):  
Nitin Bisht
Keyword(s):  
Regular Element ◽  
Nilpotent Element ◽  
Idempotent Element ◽  
Von Neumann ◽  
Regular Elements ◽  
Basic Properties ◽  
Euclidean Domain ◽  
Von Neumann Regular

An element of a ring [Formula: see text] is said to be [Formula: see text]-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring [Formula: see text] are [Formula: see text]-precious, then [Formula: see text] is called an [Formula: see text]-precious ring. We study some basic properties of [Formula: see text]-precious rings. We also characterize von Neumann regular elements in [Formula: see text] when [Formula: see text] is a Euclidean domain and by this argument, we produce elements that are [Formula: see text]-precious but either not [Formula: see text]-clean or not precious.

PURE-INJECTIVITY FROM A DIFFERENT PERSPECTIVE

2017 ◽  
Vol 60 (1) ◽  
pp. 135-151 ◽  
Author(s):  
S. R. LÓPEZ-PERMOUTH ◽  
J. MASTROMATTEO ◽  
Y. TOLOOEI ◽  
B. UNGOR
Keyword(s):  
Point Of View ◽  
Endomorphism Rings ◽  
Von Neumann ◽  
Basic Properties ◽  
Injective Modules ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Pure Extension ◽  
Regular Rings

AbstractThe study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

A CLOSURE OPERATION IN RINGS

2001 ◽  
Vol 12 (07) ◽  
pp. 791-812
Author(s):  
PERE ARA ◽  
GERT K. PEDERSEN ◽  
FRANCESC PERERA
Keyword(s):  
Regular Element ◽  
Closure Operation ◽  
Von Neumann ◽  
Regular Elements ◽  
C Algebras ◽  
Partial Inverse ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Invertible Elements ◽  
Maximal Regular Ideal

We study the operation E → cl (E) defined on subsets E of a unital ring R, where x ∈ cl (E) if (x + Rb) ∩ E ≠ ∅ for each b in R such that Rx + Rb = R. This operation, which strongly resembles a closure, originates in algebraic K-theory. For any left ideal L we show that cl (L) equals the intersection of the maximal left ideals of R containing L. Moreover, cl (Re) = Re + rad (R) if e is an idempotent in R, and cl (I) = I for a two-sided ideal I precisely when I is semi-primitive in R (i.e. rad (R/I) = 0). We then explore a special class of von Neumann regular elements in R, called persistently regular and characterized by forming an "open" subset Rpr in R, i.e. cl (R\Rpr) = R\Rpr. In fact, R\Rpr = cl (R\Rr), so that Rpr is the "algebraic interior" of the set Rr of regular elements. We show that a regular element x with partial inverse y is persistently regular, if and only if the skew corner (1 - xy)R(1 - yx) is contained in Rr. If I reg (R) denotes the maximal regular ideal in R and [Formula: see text] the set of quasi-invertible elements, defined and studied in [4], we prove that [Formula: see text]. Specializing to C*-algebras we prove that cl (E) coincides with the norm closure of E, when E is one of the five interesting sets R-1, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], and that Rpr coincides with the topological interior of Rr. We also show that the operation cl respects boundedness, self-adjointness and positivity.

GENERALIZATIONS OF COMPLEMENTED RINGS WITH APPLICATIONS TO RINGS OF FUNCTIONS

2009 ◽  
Vol 08 (01) ◽  
pp. 17-40 ◽  
Author(s):  
M. L. KNOX ◽  
R. LEVY ◽  
W. WM. MCGOVERN ◽  
J. SHAPIRO
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Regular Element ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Ring Of Quotients

It is well known that a commutative ring R is complemented (that is, given a ∈ R there exists b ∈ R such that ab = 0 and a + b is a regular element) if and only if the total ring of quotients of R is von Neumann regular. We consider generalizations of the notion of a complemented ring and their implications for the total ring of quotients. We then look at the specific case when the ring is a ring of continuous real-valued functions on a topological space.

Von Neumann Regular and Related Elements in Commutative Rings

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1017-1040 ◽  
Author(s):  
David F. Anderson ◽  
Ayman Badawi
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Von Neumann ◽  
Regular Elements ◽  
Zero Divisor Graph ◽  
Von Neumann Regular ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, π-regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph Γ(R) of R induced by the above elements.

scholarly journals On strongly pi-regular rings of stable range one

1995 ◽  
Vol 51 (3) ◽  
pp. 433-437 ◽  
Author(s):  
Hua-Ping Yu ◽  
Victor P. Camilo
Keyword(s):  
Regular Element ◽  
Regular Ring ◽  
Associative Ring ◽  
Stable Range ◽  
Von Neumann ◽  
Stable Range One ◽  
Von Neumann Regular ◽  
Open Question ◽  
Regular Rings

An associative ring R is said to have stable range one if for any a, b ∈ R satisfying aR + bR = R, there exists y ∈ R such that a + by is right (equivalently, left) invertible. Call a ring R strongly π-regular if for every element a ∈ R there exist a number n (depending on a) and an element x ∈ R such that an = an+1x. It is an open question whether all strongly π-regular rings have stable range one. The purpose of this note is to prove the following Theorem: If R is a strongly π-regular ring with the property that all powers of every nilpotent von Neumann regular element are von Neumann regular in R, then R has stable range one.

SUMS OF UNITS IN RINGS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350072 ◽  
Author(s):  
HARPREET K. GROVER ◽  
ZHOU WANG ◽  
DINESH KHURANA ◽  
JIANLONG CHEN ◽  
T. Y. LAM
Keyword(s):  
Regular Element ◽  
Regular Ring ◽  
Stable Range ◽  
Von Neumann ◽  
Stable Range One ◽  
Von Neumann Regular ◽  
Sums Of Units

In this paper, we study rings that are additively generated by units. We prove that if the identity in a ring with stable range one is a sum of two units, then every (von Neumann) regular element is a sum of two units. It follows that every element in a unit-regular ring is a sum of two units if the identity is a sum of two units. Also, if the identity of a strongly π-regular ring is a sum of two units, then every element is a sum of three units.

scholarly journals Pseudo-Von Neumann Regular Graph of Commutative Ring

2021 ◽  
Vol 1879 (3) ◽  
pp. 032012
Author(s):  
Nabeel E. Arif ◽  
Roslan Hasani ◽  
Nermen J. Khalel
Keyword(s):  
Commutative Ring ◽  
Regular Graph ◽  
Von Neumann ◽  
Von Neumann Regular

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.

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