The notions of the G-S inverse and the C-S inverse are extended from the set of all complex n × n matrices to *-rings. We present new characterizations of these inverses, study their properties, and apply the C-S inverse to introduce two relations on the set R⏧d of all core-EP invertible elements in a *-ring with identity. We explore the connection between these two relations and the star partial order and prove that one of them is a partial order on R⏧d.
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Marovt et al. (2024) studied this question.
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