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.

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.

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.

scholarly journals On the Geometry of the Unit Ball of aJB*-Triple

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Haifa M. Tahlawi ◽  
Akhlaq A. Siddiqui ◽  
Fatmah B. Jamjoom
Keyword(s):  
Unit Ball ◽  
Structural Properties ◽  
Extreme Points ◽  
Von Neumann ◽  
Regular Elements ◽  
Von Neumann Regular ◽  
Invertible Elements

We explore aJB*-triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting ofC*-algebras. This class of BP-quasi invertible elements properly includes all invertible elements and all extreme points of the unit ball and is properly included in von Neumann regular elements in aJB*-triple; this indicates their structural richness. We initiate a study of the unit ball of aJB*-triple investigating some structural properties of the BP-quasi invertible elements; here and in sequent papers, we show that various results on unitary convex decompositions and regular approximations can be extended to the setting of BP-quasi invertible elements. SomeC*-algebra andJB*-algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended toJB*-triples.

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.

On von Neumann regular elements in f-rings

2017 ◽  
Vol 78 (1) ◽  
pp. 119-124 ◽  
Author(s):  
Youssef Azouzi ◽  
Mohamed Amine Ben Amor
Keyword(s):  
Von Neumann ◽  
Regular Elements ◽  
Von Neumann Regular

Some types of ring elements in Ore extensions over noncommutative rings

2017 ◽  
Vol 16 (11) ◽  
pp. 1750201 ◽  
Author(s):  
E. Hashemi ◽  
M. Hamidizadeh ◽  
A. Alhevaz
Keyword(s):  
Ore Extension ◽  
Von Neumann ◽  
Noncommutative Rings ◽  
Regular Elements ◽  
Derivation Formula ◽  
Von Neumann Regular ◽  
Ore Extensions ◽  
Base Ring ◽  
Extension Ring ◽  
Ring Elements

Let [Formula: see text] be an associative unital ring with an endomorphism [Formula: see text] and [Formula: see text]-derivation [Formula: see text]. Some types of ring elements such as the units and the idempotents play distinguished roles in noncommutative ring theory, and will play a central role in this work. In fact, we are interested to study the unit elements, the idempotent elements, the von Neumann regular elements, the [Formula: see text]-regular elements and also the von Neumann local elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible. As an application, we completely characterize the clean elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible.

Regular elements in generalized hermitian algebras

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
David Foulis ◽  
Sylvia Pulmannová
Keyword(s):  
Spectral Order ◽  
Type Representation ◽  
Von Neumann ◽  
Regular Elements ◽  
Unit Space ◽  
Special Jordan Algebra ◽  
Projection Lattice ◽  
Von Neumann Regular ◽  
Algebra Relate ◽  
Order Unit Space

AbstractA generalized Hermitian (GH) algebra is a special Jordan algebra that is at the same time a spectral order-unit space. In this paper we characterize the von Neumann regular elements in a GH-algebra, relate maximal pairwise commuting subsets of the algebra to blocks in its projection lattice, and prove a Gelfand-Naimark type representation theorem for commutative GH-algebras.

scholarly journals Regular elements determined by generalized inverses

2019 ◽  
Vol 18 (07) ◽  
pp. 1950128 ◽  
Author(s):  
Adel Alahmadi ◽  
S. K. Jain ◽  
André Leroy
Keyword(s):  
Regular Ring ◽  
Semiprime Ring ◽  
Generalized Inverses ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Regular Elements ◽  
Von Neumann Regular

In a semiprime ring, von Neumann regular elements are determined by their inner inverses. In particular, for elements [Formula: see text] of a von Neumann regular ring [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] denotes the set of inner inverses of [Formula: see text]. We also prove that, in a semiprime ring, the same is true for reflexive inverses.

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.

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