Properties of core-EP order in rings with involution

We study properties of a relation in *-rings, called the core-EP (pre)order which was introduced by H. Wang on the set of all n x n complex matrices [Linear Algebra Appl., 2016, 508: 289-300] and has been recently generalized by Y. Gao, J. Chen, and Y. Ke to *-rings [Filomat, 2018, 32: 3073-3085]. We present new characterizations of the core-EP order in *-rings with identity and introduce the notions of the dual core-EP decomposition and the dual core-EP order in *-rings.