Characterizations of core and dual core inverses in rings with involution

Abstract

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 a ∈ R and n 1, then a is core invertible if and only if there exists a Hermitian element (or a projection) p such that pa = 0, a n + p is invertible. As a consequence, a is an EP element if and only if there exists a Hermitian element (or a projection) p such that pa = ap = 0, a n + p is invertible. We also get a new characterization for both core invertible and dual core invertible of a regular element by units, and their expressions are shown. In particular, we prove that for n 2, a is both Moore-Penrose invertible and group invertible if and only if (a * ) n is invertible along a.

Right core inverse and the related generalized inverses

“…p ∈ R is a projection if p 2 = p = p * . In [8], Li and Chen gave the characterizations and expressions of core inverse of an element by projections and units.…”

“…In this article, we will consider the case that a n + p is one-sided invertible, and then the concepts of (generalized) one-sided core inverses are introduced. It is worth mentioning that in [8] the authors proved that a ∈ R(a * ) n a is equivalent to a ∈ Ra * a ∩ a n R for n ≥ 1. In this article, we will prove that this condition is also a characterization of right core inverse of a.…”

Core and dual core inverses of a sum of morphisms

“…There are some papers characterizing the core and dual core inverse by units. (See for example, [12] and [11].) Inspired by them and the above two lemmas, we get characterizations of the core invertibility of a morphism with kernel.…”

The core and dual core inverse of a morphism with factorization

“…In [14] and [15], authors investigated the coexistence of core inverse and dual core inverse of an element in a * -ring which is a ring with an involution * provided that there is an anti-isomorphism * such that (a * ) * = a, (a+b) * = a * + b * and (ab) * = b * a * for all a, b ∈ R. It makes sense to investigate the coexistence of core inverse and dual core inverse of an R-morphism in the category of R-modules of a given ring R. We give some characterizations about the coexistence of core inverse and dual core inverse of an R-morphism in the third part.…”