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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