In this paper, double commutativity and the reverse order law for the core inverse are considered. Then, new characterizations of the Moore-Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered.

Characterizations and representations of core and dual core inverses

In this paper, double commutativity and the reverse order law for the core inverse are considered. Then new characterizations of the Moore–Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered.

https://repositorium.sdum.uminho.pt/bitstream/1822/47036/1/Characterizations%20and%20representations%20of%20core%20and%20dual%20core%20inverses%20in%20rings.pdf

In this paper, we introduce a new notion in a semigroup as an extension of Mary’s inverse. Let . An element is called left (resp. right) invertible along if there exists such that (resp. ) and (resp. ). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) -regularity and left (right) -regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally, we study the (left, right) inverse along a product in a ring, and, as an application, Mary’s inverse along a matrix is expressed.

/pdf/further-results-on-the-inverse-along-an-element-in-3iofyekkrc.pdf

Further results on the inverse along an element in semigroups and rings

Left and right generalized inverses

Let R be a unital ring with involution. We give the characterizations and representations of the core and dual core inverses of an element in R by Hermitian elements (or projections) and units. For example, let and . Then a is core invertible if and only if there exists a Hermitian element (or a projection) p such that and is invertible. As a consequence, a is an element if and only if there exists a Hermitian element (or a projection) p such that and is invertible. We also get a new characterization for both core invertibility and dual core invertibility of a regular element by units, and their expressions are shown. In particular, we prove that for , a is both Moore–Penrose invertible and group invertible if and only if is invertible along a.

Characterizations of core and dual core inverses in rings with involution

Generalized inverses of a factorization in a ring with involution

In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix.

https://repositorium.sdum.uminho.pt/bitstream/1822/39182/1/The%20inverse%20along%20a%20product%20and%20its%20applications.pdf

The inverse along a product and its applications

In this paper, we investigate the pseudo Drazin invertibility of the sum and the product of elements in a Banach algebra $\mathscr{A}$. Given pseudo Drazin invertible elements $a$ and $b$ such that $a^2b=aba$ and $b^2a=bab$, it is shown that $ab$ is pseudo Drazin invertible and $a+b$ is pseudo Drazin invertible if and only if so is $1+a^ \ddagger b$, and the related formulae are provided.

/pdf/representations-for-the-pseudo-drazin-inverse-of-elements-in-fssii440hb.pdf

Huihui Zhu

Papers

Further results on the inverse along an element in semigroups and rings

Generalized inverses of a factorization in a ring with involution

The inverse along a product and its applications

Representations for the pseudo drazin inverse of elements in a banach algebra

Centralizers and their applications to generalized inverses