  Given a ring $R$, we study the bimodules $M$ for which the trivial extension
$R\propto M$ is morphic. We obtain a complete characterization in the case
where $R$ is left perfect, and we prove that $R\propto Q/R$ is morphic when $R$
is a commutative reduced ring with classical ring of quotients $Q$. We also
extend some known results concerning the connection between morphic rings and
unit regular rings.
