Prime Ideals in Noetherian Rings: A Survey

Abstract

Abstract. We consider the structure of the partially ordered set of prime ideals in a Noetherian ring. The main focus is Noetherian two-dimensional integral domains that are rings of polynomials or power series. IntroductionThe authors have been captivated by the partially ordered set of prime ideals for about four decades. Their initial motivation was interest in Kaplansky's question, Key words and phrases. prime ideals, Noetherian. The authors thank the University of Nebraska for its support of our research projects.

Open Problems in Commutative Ring Theory

“…This question is close to Question 2.15.1 of [110]. It is also observed in Example 2.14 of [110], and Corollary 7 of [108], that, if L is an algebraically closed …eld of in…nite transcendence degree over Q, then the spectra of L[x; y; z]=(x 4 + y 4 + z 4 1) and L[x; y] are known to be non-isomorphic.…”

“…Moreover, the poset Spec(Z[y]) is characterized, among posets, by …ve speci…c axioms [21,Theorem 2.9]. For more background information, see references [21,108,109,110]. Problem 40c.…”

“…If Spec(F [x; y] and Spec(K[x; y]) are orderisomorphic, are F and K necessarily isomorphic …elds? This is Question 2 of [110].…”

Minimal Zero-Dimensional Extensions of Rings of Dimension Greater than One

“…We observe that we cannot apply the main result to Noetherian rings of Krull dimension greater than two, because, for a Noetherian ring, if there is a prime ideal strictly between two given prime ideals, then there are infinitely many (see [9]). …”

“…For condition 2, the ideal is zero; for condition 3, the ideal is A + B where A and B are the prime ideals generating the two principal upsets. (These conditions are three of the conditions on the partially ordered set of prime ideals of a Noetherian ring in [9]). …”