Additive Property of Drazin Invertibility of Elements

In this article, we investigate additive properties of the Drazin inverse of elements in rings and algebras over an arbitrary field. Under the weakly commutative condition of $ab = \lambda ba$, we show that $a-b$ is Drazin invertible if and only if $aa^{D}(a-b)bb^{D}$ is Drazin invertible. Next, we give explicit representations of $(a+b)^{D}$, as a function of $a, b, a^{D}$ and $b^{D}$, under the conditions $a^{3}b = ba$ and $b^{3}a = ab$.Comment: 17 page

Reverse order law for the inverse along an element

In this paper, we introduce a new concept called left (right) g-MP inverse in a *-monoid. The relations of this type of generalized inverse with left inverse along an element are investigated. Also, the reverse order law for the inverse along an element is studied. Then, the existence criteria and formulae of the inverse of the product of triple elements along an element are investigated in a monoid. Finally, we further study left and right g-MP inverses, the inverse along an element in the context of rings.- This research is 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]; the Scientific Innovation Research of College Graduates in Jiangsu Province [No. CXLX13-072]; the Scientific Research Foundation of Graduate School of Southeast University, the FEDER Funds through 'Programa Operacional Factores de Competitividade-COMPETE', the Portuguese Funds through FCT - 'Fundacao para a Ciencia e a Tecnologia', within the project [UID/MAT/00013/2013].info:eu-repo/semantics/publishedVersio

The Moore-Penrose inverse of differences and products of projectors in a ring with involution

In this paper, we study the Moore–Penrose inverses of differences and products of projectors in a ring with involution. Some necessary and sufficient conditions for the existence of the Moore–Penrose inverse are given. Moreover, the expressions of the Moore–Penrose inverses of differences and products of projectors are presented.Portuguese Funds through FCT - ‘Fundação para a Ciência e Tecnologia’, within the project PEst-OE/MAT/UI0013/2014.info:eu-repo/semantics/publishedVersio

Further results on the inverse along an element in semigroups and rings

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