*-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{i}cio 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 Wangs 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 $aa^{scriptsizetextcircled{tiny D}}=bb^{scriptsizetextcircled{tiny D}}$, which contribute to investigate *-DMP elements.