Standard bases in mixed power series and polynomial rings over rings (Q321289)

The authors study standard bases for submodules of a mixed power series and polynomial ring \(R[[t_1,\dots,t_m]][x_1,\dots, x_n]^s\) 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 \(n = 0\) they get standard bases for polynomial rings respectively for power series rings over \(R\) as a special case. This paper should be seen as a unified approach for the existence of standard bases in a mixed power series and polynomial rings for coefficient rings which are not fields.