Pedro Patrício

Elements of rings with equal spectral idempotents

In this paper we define and study a generalized Drazin inverse x D for ring elements x, and give a characterization of elements a, b for which aa D = bb D . We apply our results to the study of EPelements of a ring with involution.

Some additive results on Drazin inverses

In this paper, some additive results on Drazin inverse of a sum of Drazin invertible elements are derived. Some converse results are also presented.

Generalized inverses modulo ℋ in semigroups and rings

The definition of the inverse along an element was very recently introduced, and it contains known generalized inverses such as the group, Drazin and Moore–Penrose inverses. In this article, we first prove a simple existence criterion for this inverse in a semigroup, and then relate the existence of such an inverse in a ring to the ring units.

About the von Neumann regularity of triangular block matrices.

Abstract Necessary and sufficient conditions are given for the von Neumann regularity of triangular block matrices with von Neumann regular diagonal blocks over arbitrary rings. This leads to the characterization of the von Neumann regularity of a class of triangular Toeplitz matrices over arbitrary rings. Some special results and a new algorithm are derived for triangular Toeplitz matrices over commutative rings. Finally, the Drazin invertibility of some companion matrices over arbitrary rings is considered, as an application.

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

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.

Moore–Penrose invertibility in involutory rings: the case aa †=bb †

In this article, we consider Moore–Penrose invertibility in rings with a general involution. Given two von Neumann regular elements a, b in a general ring with an arbitrary involution, we aim to give necessary and sufficient conditions to aa † = bb †. As a special case, EP elements are considered.

The Moore-Penrose inverse of von Neumann regular matrices over a ring

Abstract Necessary and sufficient conditions are given in order that a von Neumann regular matrix A over an arbitrary ring, and also the matrices arising in certain factorizations of A be Moore–Penrose invertible.

GENERALIZED INVERSES OF A SUM IN RINGS

We study properties of the Drazin index of regular elements in a ring with a unity 1. We give expressions for generalized inverses of 1+ba in terms of generalized inverses of 1+ab. In our development we prove that the Drazin index of 1 + ba is equal to the Drazin index of 1 + ab. doi:10.1017/S0004972710000080

On the Drazin index of regular elements

It is known that the existence of the group inverse a# of a ring element a is equivalent to the invertibility of a2a− + 1 − aa−, independently of the choice of the von Neumann inverse a− of a. In this paper, we relate the Drazin index of a to the Drazin index of a2a− + 1 − aa−. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks.

The diamond partial order in rings

Abstract In this paper we introduce a new partial order on a ring, namely the diamond partial order. This order is an extension of a partial order defined in a matrix setting in [J.K. Baksalary and J. Hauke, A further algebraic version of Cochran’s theorem and matrix partial orderings, Linear Algebra and its Applications, 127, 157–169, 1990]. We characterize the diamond partial order on rings and study its relationships with other partial orders known in the literature. We also analyse successors, predecessors and maximal elements under the diamond order.