On some classes of integral domains defined by Krullʼs a.b. operations

Abstract

Let D be an integral domain with quotient field K. The boperation that associates to each nonzero D-submodule E of K, E b := {EV | V valuation overring of D}, is a semistar operation that plays an important role in many questions of ring theory (e.g., if I is a nonzero ideal in D, I b coincides with its integral closure). In a first part of the paper, we study the integral domains that are b-Noetherian (i.e., such that, for each nonzero ideal I of D, I b = J b for some a finitely generated ideal J of D). For instance, we prove that a b-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily b-Noetherian. We also characterize several distinguished classes of b-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type ⋆a canonically associated to a given semistar operation ⋆ (for instance, the b-operation is the e.a.b. semistar operation of finite type canonically associated to the identity operation). These operations, introduced and studied by Krull, Jaffard, Gilmer and Halter-Koch, play a very important role in the recent generalizations of the Kronecker function ring. In particular, in the present paper, we classify several classes of integral domains having some of the fundamental operations d, t, w and v equal to some of the canonically associated e.a.b. operations b, ta, wa and va.

Spectral spaces of semistar operations

“…It is obvious that (iii) ⇒ (ii). For the reverse implication, it is enough to note that d T = b T is equivalent to T being Prüfer [21,Lemma 2].…”

$$v_c$$-Noetherian domains and Krull domains

“…star operations of finite character. Fontana-Picozza [6] studied some classes of integral domains defined by the b-operation (in the more general setting of semistar operations). For example, they showed that if D is integrally closed, then This paper consists of four sections including introduction.…”