On finiteness of prime cones over simple ADE-singularities of dimension one

ABSTRACT

A local ring is called a local ring of simple singularity of dimension one over the real field if it is isomorphic to a ring of the form ℝ{x,y}∕(f) and the number of proper ideals I of ℝ{x,y} with f∈I² is finite. We first give a complete classification of local rings of simple singularity of dimension one over the real field. We also show that the rings have an infinitely generated prime cone unless they are isomorphic to either $ℝ\{x,y\}∕(x^2\pm y^{2n})$ or $ℝ\{x,y\}∕(x^{2}y\pm y^{2n+3})$, where n is a positive integer.

KEYWORDS:

• Quadratic modules
• prime cones
• simple singularities

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13J30
• 13J05
• 14P99