  In this paper we study standard bases for submodules of a mixed power series
and polynomial ring $R[[t_1,\ldots,t_m]][x_1,\ldots,x_n]^s$ respectively of
their localization with respect to a $t$-local monomial ordering for a certain
class of noetherian rings $R$. The main steps are to prove the existence of a
division with remainder generalizing and combining the division theorems of
Grauert--Hironaka and Mora and to generalize the Buchberger criterion.
Everything else then translates naturally. Setting either $m=0$ or $n=0$ we get
standard bases for polynomial rings respectively for power series rings over
$R$ as a special case.
