  In this paper, we study Dorroh extensions of bialgebras and Hopf algebras.
Let $(H,I)$ be both a Dorroh pair of algebras and a Dorroh pair of coalgebras.
We give necessary and sufficient conditions for $H\ltimes_dI$ to be a bialgebra
and a Hopf algebra, respectively. We also describe all ideals of Dorroh
extensions of algebras and subcoalgebras of Dorroh extensions of coalgebras and
compute these ideals and subcoalgebras for some concrete examples.
