A Description of Ideals in Noetherian Rings

Abstract

For any commutative ring R there is a theory T(R) in a propositional language whose models are exactly the ideals of R. These ideals form a closed subspace of the Tychonov product topology on 2 ^R and we introduce an ultraproduct of sets operation on 2 ^R which allows us to “compute” limits in this space. The property that R is Noetherian is equivalent to the condition that any ultraproduct of its ideals is equal to an intersection of some of these ideals over some ultrafilter set. For any countable Noetherian ring R,this topological space is homeomorphic to a countable successor ordinal with its interval topology. We describe how the ideals of a Noetherian ring R are related to each other in terms of limits in this space as well as infinite intersections. Another description of these ideals is given in terms of their complete theories and their Cantor rank. This method works for any extension of T(R) and so there is a similar analysis for the set of prime ideals of R. Any non-empty closed set in the space Spec(R) has finitely many minimal elements of maximum Cantor rank. For countable Noetherian R of finite Krull dimension k ≥ 1,Spec(R) is homeomorphic to either ω ^k  · j + 1 or ω ^k−1 · j + 1 for some j < ω. For the prime ideals of Z[x ₁,…,x _k ] an explicit representation of each ideal of Krull rank 1 is given as a limit or as intersections of its ideals of Krull rank 2.

Key Words:

• Noetherian
• Patch topology
• Ultrafilter

1991 Mathematics Subject Classification:

• Primary 130C5,13E05
• Secondary 13L05,13P10

Acknowledgments