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.

Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in R. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary * -ring case by Rakić, Dinčić and Djordjević in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index in R. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of * -ring. Some equivalent characterizations for elements in R to be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited.

The core inverse for a complex matrix was introduced by Baksalary and Trenkler. Rakić, Dinčić and Djordjević generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible, in this paper, we will answer this question. We will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R, then a ∈ R # with a # = b if and only if (ab) * = ab, ba 2 = a and ab 2 = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.

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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.