Rota–Baxter coalgebras and Rota–Baxter bialgebras

We give a construction of Rota–Baxter coalgebras from Hopf module coalgebras and also derive the structures of the pre-Lie coalgebras via Rota–Baxter coalgebras of different weight. Finally, the notion of Rota–Baxter bialgebra is introduced and some examples are provided.