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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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Let H be a finite-dimensional cocommutative semisimple Hopf algebra and A * H a twisted smash product. The Calabi–Yau (CY) property of twisted smash product is discussed. It is shown that if A is a CY algebra of dimension dA, a necessary and sufficient condition for A * H to be a CY Hopf algebra is given.

Let [Formula: see text] be a semigroup and [Formula: see text]. The concept of [Formula: see text]-inverses was introduced by Drazin in 2012. It is well known that the Moore–Penrose inverse, the Drazin inverse, the Bott–Duffin inverse, the inverse along an element, the core inverse and dual core inverse are all special cases of the [Formula: see text]-inverse. In this paper, a new relationship between the [Formula: see text]-inverse and the Bott–Duffin [Formula: see text]-inverse is established. The relations between the [Formula: see text]-inverse of [Formula: see text] and certain classes of generalized inverses of [Formula: see text] and [Formula: see text], and the [Formula: see text]-inverse of [Formula: see text] are characterized for some [Formula: see text], where [Formula: see text]. Necessary and sufficient conditions for the existence of the [Formula: see text]-inverse of a lower triangular matrix over an associative ring [Formula: see text] are also given, and its expression is derived, where [Formula: see text] are regular triangular matrices.

Let R be a ring and b, c ∈ R. The concept of (b, c)-inverses was introduced by Drazin in 2012. In this paper, the existence and the expression of the (b, c)-inverse in a ring with an involution are investigated.