On the Covariance of Moore-Penrose Inverses in Rings with Involution

Abstract and Applied Analysis ◽

10.1155/2014/309708 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Hesam Mahzoon

Keyword(s):

Regular Element ◽

Relative Topology ◽

Rings With Involution ◽

Invertible Elements

We consider the so-called covariance set of Moore-Penrose inverses in rings with an involution. We deduce some new results concerning covariance set. We will show that ifais a regular element in aC∗-algebra, then the covariance set ofais closed in the set of invertible elements (with relative topology) ofC∗-algebra and is a cone in theC∗-algebra.

Download Full-text

Core partial order in rings with involution

Filomat ◽

10.2298/fil1718695z ◽

2017 ◽

Vol 31 (18) ◽

pp. 5695-5701 ◽

Cited By ~ 2

Author(s):

Xiaoxiang Zhang ◽

Sanzhang Xu ◽

Jianlong Chen

Keyword(s):

Partial Order ◽

Partial Orders ◽

Reverse Order ◽

Unital Ring ◽

Reverse Order Law ◽

Rings With Involution ◽

Invertible Elements

Let R be a unital ring with involution. Several characterizations and properties of core partial order in R are given. In particular, we investigate the reverse order law (ab)# = b#a# for two core invertible elements a, b ? R. Some relationships between core partial order and other partial orders are obtained.

Download Full-text

A CLOSURE OPERATION IN RINGS

International Journal of Mathematics ◽

10.1142/s0129167x01001039 ◽

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.

Download Full-text

Further results of special elements in rings with involution

Filomat ◽

10.2298/fil2010381m ◽

2020 ◽

Vol 34 (10) ◽

pp. 3381-3393

Author(s):

Dijana Mosic ◽

Sanzhang Xu ◽

Julio Benítez

Keyword(s):

Partial Isometries ◽

Rings With Involution ◽

Invertible Elements

In this paper, we consider Moore-Penrose invertible, group invertible, and core invertible elements in rings with involution to characterize EP, generalized normal, generalized Hermitian elements and generalized partial isometries. As a consequence, we obtain new characterizations for elements in rings with involution to be normal and Hermitian elements.

Download Full-text

On some orders in ∗-rings based on the core-EP decomposition

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500104 ◽

2020 ◽

pp. 2250010

Author(s):

Janko Marovt ◽

Dijana Mosić

Keyword(s):

Partial Order ◽

The Core ◽

Unital Ring ◽

Rings With Involution ◽

Invertible Elements ◽

Unital Rings ◽

Minus Partial Order

We study certain relations in unital rings with involution that are derived from the core-EP decomposition. The notion of the WG pre-order and the C-E partial order is extended from [Formula: see text], the set of all [Formula: see text] matrices over [Formula: see text], to the set [Formula: see text] of all core-EP invertible elements in an arbitrary unital ring [Formula: see text] with involution. A new partial order is introduced on [Formula: see text] by combining the WG pre-order and the well known minus partial order, and a new characterization of the core-EP pre-order in unital proper ∗-rings is presented. Properties of these relations are investigated and some known results are thus generalized.

Download Full-text