*-DMP elements in $*$-semigroups and $*$-rings

Abstract: 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.