*-DMP Elements in *-Semigroups and *-Rings

In this paper, we investigate *-DMP elements in *-semigroups and *-rings. The notion of *DMP element was introduced by Patr ' icio and Puystjens in 2004. An element a is *-DMP if there exists a positive integer m such that am is EP. We first characterize *-DMP elements in terms of the f 1,3 g -inverse, Drazin inverse and pseudo core inverse, respectively. Then, we characterize the core-EP decomposition utilizing the pseudo core inverse, which extends the core-EP decomposition introduced by Wang for complex matrices to an arbitrary *-ring; and this decomposition turns to be a useful tool to characterize *-DMP elements. Further, we extend Wang's core-EP order from complex matrices to *-rings and use it to investigate *-DMP elements. Finally, we give necessary and su ffi cient conditions for two elements a; b in *-rings to have aa (sic) = bb (sic), which contribute to study *-DMP elements.