Standard bases in mixed power series and polynomial rings over rings

In this paper we study standard bases for submodules of a mixed power series and polynomial ring R ź t 1 , ź , t m ź x 1 , ź , x n s respectively of their localisation with respect to a t _ -local monomial ordering for a certain class of noetherian rings R, also called Zacharias rings. The main steps are to prove the existence of a division with remainder generalising and combining the division theorems of Grauert-Hironaka and Mora and to generalise 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.