Pseudo core invertibility and DMP invertibility in two semigroups of a ring with involution

. In 2004, Patr´ıcio and Puystjens characterized the relation between Drazin invertible elements (resp., Moore-Penrose invertible elements) of two semigroups pRp and pRp + 1 − p of a ring R for some idempotent (resp., projection) p ∈ R . In this paper, we consider the relevant result for pseudo core invertible elements of such two semigroups of a ring for some projection, which is then applied to characterize the relation between pseudo core invertible elements of the matrix semigroup AA † R m × m AA † + I m − AA † and the matrix semigroup A † AR n × n A † A + I n − A † A , where A ∈ R m × n with A † existing. Also, similar equivalence involving DMP invertible elements is investigated.