Group, Moore–Penrose, core and dual core inverse in rings with involution

Star, sharp, core and dual core partial order in rings with involution

“…rankðAÞ ¼ rankðA 2 Þ in which case they are unique. In [17] authors considered the MP, group, core and dual core inverses in the setting of arbitrary Ã-ring. It is shown that X 2 M n is core inverse of A 2 M n if and only if X satisfies Eqs.…”

“…Because of that, if there exists a 2 af1; 2; 3; 6; 7g then a is called the core inverse of a. In Theorem 2.14 in [17] it is shown that a is core inverse of a if and only if axa ¼ a; a R ¼ aR and Ra ¼ Ra à in which case a is unique. Similarly, if there exists a 2 af1; 2; 4; 8; 9g; then a is unique and it is called the dual core inverse of a.…”

“…Let R ð1Þ ; R y ; R # ; R ; R denote the set of all regular, MP, group, core, dual core invertible elements of a ring R respectively. It is shown in [17] that…”

“…By Lemma 3.1 and representation for a given in(16), it follows that x 2 af1; x ij are arbitrary elements in appropriate subsets of R. Note that e 1 þ e 2 ¼ bb and f 1 þ f 2 ¼ b b. By Lemma 3.1 and representation for b given in(17), it follows that x 2 bf1; 3g if and only if…”

Partial orders in rings based on generalized inverses – unified theory

“…An element a ∈ R has dual core inverse if there exists an x ∈ R such that the following equations hold [28]:…”