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[EN] Let A be an complex matrix. The (B,C)-inverse A vertical bar(B,C) of A was introduced by Drazin in 2012. For given matrices A and B, several rank equalities related to A vertical bar(B1,C1) and B vertical bar(B2,C2) of A and B are presented. As applications, several rank equalities related to the inverse along an element, the Moore-Penrose inverse, the Drazin inverse, the group inverse and the core inverse are obtained., The second author is grateful to China Scholarship Council for giving him a scholarship for his further study in Universitat Politecnica de Valencia, Spain.

In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix., The authors are highly grateful to the referee for valuable comments which led to improvements of the paper. In particular, Remarks 3.2 and 3.4 were suggested to the authors by the referee. The first author is grateful to China Scholarship Council for supporting him to purse his further study in University of Minho, Portugal. Pedro Patr´ıcio and Yulin Zhang were 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. Jianlong Chen and Huihui Zhu were supported 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).

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.

The concept of the inverse along an element was introduced by Mary in 2011. Later, Zhu et al. introduced the one-sided inverse along an element. In this paper, we first give a new existence criterion for the one-sided inverse along a product and characterize the existence of Moore–Penrose inverse by means of one-sided invertibility of certain element in a ring. In addition, we show that a∈ S † ⋂ S # if and only if (a∗a)k is invertible along a if and only if (aa∗)k is invertible along a in a ∗ -monoid S, where k is an arbitrary given positive integer. Finally, we prove that the inverse of a along aa ∗ coincides with the core inverse of a under the condition a∈ S { 1 , 4 } in a ∗ -monoid S., FCT - Fuel Cell Technologies Program(CXLX13-072), This research was supported by the National Natural Science Foundation of China (No. 11371089), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120092110020), the Natural Science Foundation of Jiangsu Province (No. BK20141327) and the Foundation of Graduate Innovation Program of Jiangsu Province (No. KYZZ15-0049)., info:eu-repo/semantics/publishedVersion

In this article one-sided (b, c)-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In adddition, the (b, c)-inverse and the inverse along an element will be also researched in the context of rectangular matrices., Comment: 38 pages, original research article

In this article several properties of the inverse along an element will be studied in the context of unitary rings. New characterizations of the existence of this inverse will be proved. Moreover, the set of all invertible elements along a fixed element will be fully described. Futhermore, commuting inverse along an element will be characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) will be also considered., Comment: 17 pages, original research article

In this paper, we investigate the recently defined notion of inverse along an element in the context of matrices over a ring. Precisely, we study the inverse of a matrix along a lower triangular matrix, under some conditions., This research was financed by FEDER Funds through ``Programa Operacional Factores de Competitividade - COMPETE'' and by Portuguese Funds through FCT - ``Fundação para a Ciência e a Tecnologia'', within the project PEst-C/MAT/UI0013/2011.