  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 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 $aa^{\scriptsize\textcircled{\tiny
D}}=bb^{\scriptsize\textcircled{\tiny D}}$, which contribute to investigate
*-DMP elements.
