On a weighted core inverse in a ring with involution

Idempotents generated by weighted generalized inverses in rings with involution

“…Proof. The proof of (i)-(iii) were given in [8,Theorem 2.5]. For completeness, we shall give a brief proof.…”

“…In 2004, Patrício and Puystens [11,Corollary 3] gave some characterizations about EP elements and proved that a ∈ R is EP if and only if a ∈ R # , aR = a * R if and only if a ∈ R † , aR = a * R. In 2008, Mosić et al [8,Theorem 2.4] extended the results to the case of weighted EP elements. Next, we give more characterizations of weighted EP elements in rings.…”

“…Following [8], an element a ∈ R is weighted EP with respect to (e, f ) if a ∈ R # ∩ R † e, f and a # = a † e, f . We use the symbol [a, b] = ab − ba to denote the commutator of a and b, and R −1 to denote the set of all invertible elements in R.…”

Right core inverse and the related generalized inverses

“…(i) ⇒ (ii) For q = ax, we get qa k = a k and qx = x implying (4.1). Since In [9], the authors introduced the definition of weighted core inverse, called it e-core inverse of a, where e is an invertible Hermitian element. Similarly, the concept of one-sided e-core inverse of a can be given:…”

Determinantal Representations of the Core Inverse and Its Generalizations with Applications

“…Other generalizations of the core inverse were recently introduced for n × n complex matrices, namely, BT inverses [5], DMP inverses [6], CMP inverses [7], etc. e characterizations, computing methods, some applications of the core inverse, and its generalizations were recently investigated in complex matrices and rings (see, e.g., [3,[8][9][10][11][12][13][14][15][16][17][18]).…”