  The superconducting critical temperature $T_C$ of a
superconductor/ferromagnet (S/F) bilayer with spin-flip scatterings at the
interface is calculated as a function of the ferromagnet thickness $d_F$ in the
dirty limit employing the Usadel equation. The appropriate boundary conditions
from the spin-flip scatterings at the S/F interface are derived for the Usadel
equation which includes the spin triplet pairing components as well as the spin
singlet one. The spin-flip processes induce the spin triplet pairing components
with s-wave in momentum and odd symmetry in frequency from the s-wave singlet
order parameter $\Delta$ of the superconductor region. The induced triplet
components alter the singlet order parameter in the superconductor through
boundary conditions at the interface and, consequently, change the $T_C$ of an
S/F bilayer system. The calculated $T_C(d_F)$, like the case of no spin-flips,
shows non-monotonic behavior which typically decreases as $d_F$ is increased
from 0 and shows a shallow minimum and then saturates slowly as $d_F$ is
further increased. It is well established that as the interface resistance
(parameterized in terms of $\gamma_b$) is increased, the $T_C$ is increased for
a given $d_F$ and the non-monotonic feature in $T_C(d_F)$ is strongly
suppressed. As the spin flip scattering (parameterized in terms of $\gamma_m$)
is increased, on the other hand, the $T_C$ is also increased for a given $d_F$,
but the non-monotonic feature in $T_C(d_F)$ is less suppressed or even
enhanced, through the formation of the spin triplet components.
