Prime ideals in Noetherian rings
University of Nebraska.
Let R be a commutative Noetherian ring. We consider the set Spec(R) of prime ideals of R as a partially ordered set, ordered by inclusion. Around 1950 Irving Kaplansky asked, "Which partially ordered sets arise as Spec(R) for some Noetherian ring R?" This question is completely open, even if only two-dimensional sets are considered, despite a large amount of work over the intervening years by many mathematicians, such as Hochster, Heitmann, Nagata, McAdam, and Ratliff. In this talk, we describe prime spectra for some two-dimensional rings of polynomials and power series. This involves our work and work of William Heinzer, Roger Wiegand and our students.