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Register a new userRegular elements determined by generalized inverses
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In a semiprime ring, von Neumann regular elements are determined by their inner inverses. In particular, for elements $a,b$ of a von Neumann regular ring $R$, $a=b$ if and only if $I(a)=I(b)$, where $I(x)$ denotes the set of inner inverses of $xin R$. We also prove that, in a semiprime ring, the same is true for reflexive inverses.
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In this paper, we introduce two new generalized inverses of matrices, namely, the $bra{i}{m}$-core inverse and the $pare{j}{m}$-core inverse. The $bra{i}{m}$-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler cite{BT} and the core-EP inverse defined by Manjunatha Prasad and Mohana cite{MM}. The $pare{j}{m}$-core inverse of a complex matrix extends the notions of the core inverse and the ${rm DMP}$-inverse defined by Malik and Thome cite{MT}. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers.
In this paper, we introduce the notion of a (generalized) right core inverse and give its characterizations and expressions. Then, we provide the relation schema of (one-sided) core inverses, (one-sided) pseudo core inverses and EP elements.
Let R be a unit-regular ring, and let a,b,c in R satisfy aba=aca. If ac and ba are group invertible, we prove that ac is similar to ba. Furthermore, if ac and ba are Drazin invertible, then their Drazin inverses are similar. For any ntimes n complex matrices A,B,C with ABA = ACA ,we prove that AC and BA are similar if and only if their k-powers have the same rank. These generalize the known Flanders theorem proved by Hartwig.
In this paper, we find the roots of lightlike quaternions. By introducing the concept of the Moore-Penrose inverse in split quaternions, we solve the linear equations $axb=d$, $xa=bx$ and $xa=bbar{x}$. Also we obtain necessary and sufficient conditions for two split quaternions to be similar or consimilar.
We give some statements that are equivalent to the existence of group inverses of Peirce corner matrices of a $2 times 2$ block matrix and its generalized Schur complements. As applications, several new results for the Drazin inverses of the generalized Schur complements and the $2 times 2$ block matrix are obtained and some of them generalize several results in the literature.
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