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In this paper, we investigate *-DMP elements in * -semigroups and * -rings.The notion of *-DMP element was introduced by Patrício 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 give the pseudo core decomposition utilizing the pseudo core inverse, which extends the core-EP decomposition introduced by Wang for 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 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 investigate *-DMP elements.

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