  A triangular matrix ring A is defined by a triplet (R,S,M) where R and S are
rings and M is an S-R-bimodule. In the main theorem of this paper we show that
if T is a tilting S-module, then under certain homological conditions on M as
an S-module, one can extend T to a tilting complex over A inducing a derived
equivalence between A and another triangular matrix ring specified by
(S',R,M'), where the ring S' and the R-S'-bimodule M' depend only on T and M,
and S' is derived equivalent to S. Note that no conditions on the ring R are
needed.
  These conditions are satisfied when S is an Artin algebra of finite global
dimension and M is finitely generated as an S-module. In this case,
$(S',R,M')=(S,R,DM)$ where D is the duality on the category of finitely
generated S-modules. They are also satisfied when S is arbitrary, M has a
finite projective resolution as an S-module, and $\Ext^n_S(M,S)=0$ for all
$n>0$. In this case, $(S',R,M')=(S,R,\Hom_S(M,S))$.
