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.

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.

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.

scholarly journals On regular ideals in reduced rings

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3715-3726 ◽  
Author(s):  
A.R. Aliabady ◽  
R. Mohamadian ◽  
S. Nazari
Keyword(s):  
Closed Subset ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Closed Set ◽  
Von Neumann ◽  
Arbitrary Ring ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Isolated Points ◽  
Maximal Regular Ideal

Let R be a commutative ring with identity and X be a Tychonoff space. An ideal I of R is Von Neumann regular (briefly, regular) if for every a ? I, there exists b ? R such that a = a2b. In the present paper, we obtain the general form of a regular ideal in C(X) which is OA, for some closed subset A of ?X, for which Ac?X ? (P(X))?, where P(X) is the set of all P-points of X. We show that the ideals and subrings such as CK(X), C?(X), C?(X), SocmC(X) and M?X\X are regular if and only if they are equal to the socle of C(X). We carry further the study of the maximal regular ideal, for instance, it is shown that for a vast class of topological spaces (we call them OPD-spaces) the maximal regular ideal is OX\I(X), where I(X) is the set of isolated points of X. Also, for this class, the socle of C(X) is the maximal regular ideal if and only if I(X) contains no infinite closed set. We also show that C(X) contains an ideal which is both essential and regular if and only if (P(X))? is dense in X. Finally it is shown that, for semiprimitive rings pure ideals are of the form OA which A is a closed subset of Max(R), also a P-point of X = Max(R) is introduced and it is shown that the maximal regular ideal of an arbitrary ring R is OX\P(X), which P(X) is the set of P-points of X = Max(R).

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

Properly algebraic and spectrum-finite ideals in Jordan systems

1993 ◽  
Vol 114 (1) ◽  
pp. 149-161 ◽  
Author(s):  
Ottmar Loos
Keyword(s):  
Triple System ◽  
Von Neumann ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Algebraic Elements ◽  
Algebraic Ideal ◽  
Jordan System

The two main results of this paper are:(i) The set of properly algebraic elements of a Jordan system (algebra, triple system or pair) over an uncountable field is an ideal.(ii) For a semiprimitive Banach Jordan system, the socle, the largest properly algebraic ideal, the largest properly spectrum-finite ideal and the largest von Neumann regular ideal all coincide.

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.

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.

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