In this paper, we investigate *-DMP elements in *-semigroups and *-rings. The notion of *-DMP element was introduced by Patrício and Puystjens in 2004. An element a is *-DMP if there exists a positive integer m such that a m is EP. We first characterize *-DMP elements in terms of the {1,3}-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 sufficient conditions for two elements a, b in *-rings to have aa D = bb D , which contribute to study *-DMP elements.