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 EP elements in a ring with involution.2000 Mathematics subject classification: primary 16A32, 16A28, 15A09; secondary 46H05, 46L05.

Abstract. In this paper, double commutativity and the reverse order law for the core inverse are considered. en, 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.

In this paper, we introduce a new notion in a semigroup S as an extension of Mary's inverse. Let a, d ∈ S. An element a is called left (resp. right) invertibleAn existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) π -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.

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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.2000 Mathematics subject classification: primary 15A09; secondary 16U99.

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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 analyze successors, predecessors and maximal elements under the diamond order.