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Let $R$ be a ring with involution. The recently introduced notions of thecore and dual core inverse are extended from matrix to an arbitrary $*$-ringcase. It is shown that the group, Moore-Penrose, core and dual core inverse areclosely related and they can be treated in the same manner using appropriateidempotents. The several characterizations of these inverses are given. Somenew properties are obtained and some known results are generalized. A number ofcharacterizations of EP elements in $R$ are obtained. It is shown that core anddual core inverse belong to the class of inverses along an element and to theclass of $(b,c)$-inverses. The case when $R$ is algebra of all bounded linearoperators on Hilbert space is specifically considered.

Group, Moore-Penrose, core and dual core inverse in rings with involution