Some results on simple complete ideals having one characteristic pair

  • Silvio Greco
  • Karlheinz Kiyek
Keywords: Simple complete ideals, Polar ideal, Value semigroup, Two- dimensional regular local rings, Valuation associated to a simple ideal, Minimal system of generators, Monomial ideals


Let α be a regular local two-dimensional ring, and let m = (x, y) be its maximal ideal. Let m > n > 1 be coprime integers, and let p be the integral closure of the ideal (x^m , y^n ). Then p is a simple complete m-primary ideal, and its value semigroup is generated by m, n.

We construct a minimal system of generators {z_0 , . . . , z_n } of p, and from this we get a minimal system of generators of the polar ideal p' of p, consisting of n = θ elements. In particular, we show that p and p' are monomial ideals. When α = κ[ [ x, y ] ], a ring of formal power series over an algebraically closed field κ of characteristic zero, this implies the existence of some relevant property.