Standard Bases in Power Series Rings: Uniqueness and Superfluous Critical Pairs

We show that w.r.t. fixed admissible term order, every ideal in a ring of power series over a field has a unique reduced standard basis. Furthermore, we show that a finite set of power series whose lowest terms are pairwise relatively prime is a standard basis. Finally, a second criterion for detecting superfluous critical pairs is discussed. As a spinoff, we obtain a Gröbner basis criterion that does not seem to have been stated before.