In this paper, we introduce a new notion in a semigroup $S$ as an extension of Mary's inverse. Let $a,d\in S$. An element $a$ is called left (resp. right) invertible along $d$ if there exists $b\in S$ such that $bad=d$ (resp. $dab=b$) and $b\leq_\mathcal{L}d$ (resp. $b\leq_\mathcal{R}d$). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) $\pi$-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.The authors are highly grateful to the referee for valuable comments which led to improvements of this paper. In particular, Corollaries 2.5, 2.6 and 3.6, Remarks 2.13 and 3.10 and the final remark (ii) were suggested to the authors by the referee. The first author is grateful to China Scholarship Council for giving him a purse for his further study in University of Minho, Portugal. Jianlong Chen and Huihui Zhu are financed by the National Natural Science Foundation of China (No. 11201063 and No. 11371089), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120092110020), the Natural Science Foundation of Jiangsu Province (No. BK20141327), the Foundation of Graduate Innovation Program of Jiangsu Province(No. CXLX13-072), the Scientific Research Foundation of Graduate School of Southeast University and the Fundamental Research Funds for the Central Universities (No. 22420135011). Pedro Patr´ıcio is financed by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the “Funda¸c˜ao para a Ciˆencia e a Tecnologia”, through the Project PEst-OE/MAT/UI0013/2014

Let R be a unital $\ast$-ring. In this paper, we introduce pseudo DMP inverses and generalized m-EP elements in R, which can been seen as extensions of DMP inverses and m-EP matrices, respectively. Also, we investigate both pseudo DMP inverses and generalized m-EP elements for their characterizations and properties.

Let R be a unital -ring. In this paper, we give several characterizations for the pseudo core and dual pseudo core inverses of an element by units in R. Moreover, their relations with -DMP are obtained. As a special case, the existence criteria for EP elements ( -DMP with index 1) are given. Then, the double commutant property of hybrid (b, c)-inverses is considered. Finally, the relations of the inverse along an element and the hybrid (b, c)-inverse, the (dual) pseudo core inverses and the hybrid (b, c)-inverse are investigated.

Let R be an associative ring with unity 1 and let a,b,c∈R. In this paper, several characterizations for hybrid (b,c)-inverses of a are given. Also, the hybrid (b,c)-inverse of a is characterized by the group inverse of ab, under certain hypothesis. In particular, existence criteria for the the inverse along an element are obtained. Finally, we get the double commutant property and the reverse order law of annihilator (b,c)-inverses.