Prime ideals of Moh and the characteristic of the field

TL;DR

This paper revisits Moh's result on the minimal generators of prime ideals in power series rings, generalizes it, and explores how the characteristic of the field influences the minimal number of generators, revealing new complexities.

Contribution

It provides a generalized lower bound for the minimal generators of prime ideals in three-variable power series rings and examines characteristic-dependent variations.

Findings

The minimal number of generators of P can decrease depending on the field's characteristic.

The minimal generating sets are standard bases with negative degree reverse lexicographic order.

Contradicts previous assumptions by Sally regarding the invariance of minimal generators.

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.