EP Elements Research Articles

One-sided EP elements in rings with involution

One-sided EP elements in rings with involution

The equation characterizations of generalized inverses in a ring with involution

In this paper, we study the relation between the generalized inverse properties of an element in a ring with involution and related equations. Mainly, by exploring the existence of the solution in a given set and the expressions of the general solution to these constructed related equation, we obtain a lot of new characterizations of EP elements, partial isometries, SEP elements, Hermitian elements and normal elements.

Special Sylvester equations, PI and EP elements in rings with involution

In this paper, we study the general solution to certain special Sylvester equations, PI and EP elements in rings with involution. Especially, some practical necessary and sufficient solvability conditions for special Sylvester equations [Formula: see text] and [Formula: see text] are presented, and expressions of the general solution to these two equations when they are solvable are provided.

Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution

Abstract This paper mainly gives some sufficient and necessary conditions for an element in a ring to be EP {\mathrm{EP}} , partial isometry and strongly EP {\mathrm{EP}} by some equalities, using the solutions of certain equations and constructing the invertible elements in a ring.

Weak e–symmetric rings

We show that R is a weak symmetric ring if and only if and eRe is a symmetric ring. Further, we obtain R is weak symmetric if and only if for any , implies , where is any transformation of . As an application, we show that a ring R is a left min-abel ring if and only if R is a weak symmetric ring for any . Finally, we show that the definition of symmetric ring is not left-right symmetric. Also, with the help of weak symmetric rings, we give some characterizations of EP elements.

EP elements of Z[x]/(x2+x)

In this paper, we first show that the quotient ring Z[x]/(x2 + x) is an involution-ring with the involution ? given by (a1 +a2x)* = a1 ?a2 ?a2x, where a1, a2 ? Z. Then, we determine explicitly all invertible elements, regular elements, MP-inverses, group invertible elements, EP elements and SEP elements of Z[x]/(x2 + x). Furthermore, we give a new characterization for Abel rings.

The group inverse and Moore-Penrose inverse of the product of two elements and generalized inverse in a ring with involution

In this paper, using the expression form of the group inverse and Moore-Penrose inverse for the product of two elements, we present some new characterizations of EP elements, partial isometries, SEP elements, normal elements and hermitian elements in a ring with involution.

Strongly EP elements and the solutions of equations in rings

In this paper, we present many new characterizations of strongly EP elements in rings with involution. We especially investigate the strongly EP elements by constructing certain equations and considering the solutions of equations, revealing the existence of solutions of certain equations and the general solutions of some binary equations that play a role in characterizing strongly EP elements. Proofs of relevant conclusions are also given.

New Results on EP Elements in Rings with Involution

In this paper, we mainly give characterizations of EP elements in terms of equations. In addition, a related notion named a central EP element is defined and investigated. Finally, we focus on characterizations of a generalized EP element, i.e., [Formula: see text]-DMP element.

EP elements, partial isometries and solutions of some equations

In this paper, we firstly present new charactrizations of EP elements, partial isometries. Next, we investigate the general solutions of some equations. Finally, we discuss the relation between the consistency of certain equations and SEP elements.

Characterizations of EP elements in rings with involution

In this paper, we present a number of new characterizations of EP elements in rings with involution. Some well-known results are extended to more general settings. In addition, we specially investigate the EP elements by certain equations. Proofs of relevant conclusions are also given.

Strongly EP elements in rings with involution

The purpose of this paper is to give some sufficient and necessary conditions for an element to be a strongly [Formula: see text] element through some transformations of equations.

Partial isometry and strongly EP elements

EP elements are important research objects in the ring theory. This paper mainly gives sufficient and necessary conditions for an element in a ring to be an EP element, partial isometry, and strongly EP element by using solutions of certain equations.

Generalized inverses and solutions to related equations

In this paper, some new characterizations of partial isometries and strongly EP elements are investigated. Especially, we discuss the existence of the solutions of certain equations in a given set to characterize partial isometries, strongly EP elements and so on.

A qualitative approach to investigate emergency preparedness state for the built environment in the UAE

PurposeEmergency preparedness (EP) is one of the crucial phases of the disaster management cycle for the built environment. The body of knowledge, therefore, reports on different preparedness standards adopted by developed countries such as the United Kingdom (UK), the United States of America (USA), Canada, Japan and Australia. Other countries, however, such as the United Arab Emirates (UAE) (in the absence of its preparedness framework), have long adapted the UK preparedness standards. This has called for this study to investigate the state of EP practices in the UAE to identify the limitations and challenges it has been facing during its preparedness phase when adopting the UK preparedness standards.Design/methodology/approachQualitative methods of data collection and documentation with the content analysis were adopted to identify the barriers faced by the preparedness phase of emergency management (EM) in the UAE. A Pilot study was therefore conducted to validate eight key elements of the EP phase identified from the literature. The state of EP phase and the extent to which the eight key elements of EP elements were practiced and the barriers in their implementation in the UAE were explored through interviews at federal (National Crisis and Emergency Management Authority) and local levels (local team of crisis and emergency management).FindingsThe study identified eight key elements of the EP phase and the associated barriers related to their implementation in the UAE. The barriers were ranked based on their severity by interviewing experts at both federal and local levels.Practical implicationsThis paper addresses the need to investigate the state of the EP phase, its key elements and the barriers faced during its implementation in the UAE.Originality/valueDue to the absence of any EP frameworks or systems in the UAE, this paper aims to validate the EP elements identified by adopting a qualitative approach.

Projections generated by Moore–Penrose inverses and core inverses

Let [Formula: see text] be a unital ∗-ring. As is well known, idempotents and projections can be constructed by the Moore–Penrose inverse and the core inverse of an element in [Formula: see text]. In this paper, we mainly investigate characterizations and properties of these types of idempotents and projections. Also, several results in [Hartwig and Spindelböck, Matrices for which [Formula: see text] and [Formula: see text] commute, Linear Multilinear Algebra 14 (1984) 241–256] are extended to a general ∗-ring without ∗-cancellable conditions. As applications, the characterization of EP elements is given.

Strongly EP elements in a ring with involution

In this paper, we introduce a new class of EP elements which is called strongly EP element and give some characterizations of strongly EP elements

∗-Strongly regular rings

Let [Formula: see text] be a ring with involution ∗. An element [Formula: see text] is called ∗-strongly regular if there exists a projection [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] is invertible, and [Formula: see text] is said to be ∗-strongly regular if every element of [Formula: see text] is ∗-strongly regular. We discuss the relations among strongly regular rings, ∗-strongly regular rings, regular rings and ∗-regular rings. Also, we show that an element [Formula: see text] of a ∗-ring [Formula: see text] is ∗-strongly regular if and only if [Formula: see text] is EP. Hence we finally give some characterizations of EP elements.

Right core inverse and the related generalized inverses

In this article, we investigate some properties of right core inverses. Particularly, new characterizations and expressions for right core inverses are given, using projections and {1, 3}-inverses. Also, we introduced and investigated a new generalized right core inverse which is called right pseudo core inverse. Then, we provide the relation schema of (one-sided) core inverses, (one-sided) pseudo core inverses, and EP elements.

EP Elements in Rings with Involution

This research is supported by the National Natural Science Foundation of China (No. 11771076). The first author is grateful to China Scholarship Council for giving him a purse for his further study in Universitat Politecnica de Valencia, Spain.