Prime ideals of Moh and the characteristic of the field

Abstract

We reprove and generalize a result of Moh which gives a lower bound on the minimal number of generators of an ideal in a power series ring in three variables x,y,z over a field k. As a consequence, in each characteristic of the field k, we obtain a minimal generating set for the prime ideal P of Moh corresponding to n=3. We deduce that the minimal number of generators of P might decrease depending on the characteristic of k. This contradicts a statement of Sally and leaves as an open problem to find families of prime ideals in the power series ring in the variables x,y,z with an unbounded minimal number of generators, when k has characteristic other than zero. Finally, we show that these minimal generating sets of P are standard basis with the negative degree reverse lexicographic order.